Writing Equations in Slope-Intercept Form

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Exercises 1 In science, we often look for explanations and patterns. How will the weather be tomorrow? How will this ecosystem react if we introduce this new plant? How will the population of Nepal grow in the next decade? We can test and demonstrate theories to answer these questions with modeling. We can look at the past weather patterns and create a model simulation of how the future weather will behave. We can create a model ecosystem and see how the species involved react. We can look at birth rates and death rates to create a model for how the future of Nepal will proceed. Sometimes we are able to model these types of situations with a linear function, such as predicted human height based on femur bone length or predicted arrival of migrating birds based on miles traveled per day. When we can create a model using a linear function, we call it a linear model.
Exercises 2 The slope-intercept form is, by far, the most commonly used form for linear equations because it is so easy to graph a function just by looking at it. The slope-intercept form is: y=mx+b, where m is the slope and b is the y-intercept. When we are given the slope and y-intercept, we can simply substitute them into the formula. When we are looking at a graph, we can also easily write the equation by looking for the slope and y-intercept. Let's look at an example graph.Looking at this linear function, we can see that it crosses the y-axis at (0,1) which means that the y-intercept is 1. We can also see that as the function moves 1 step up, it moves 3 steps to the right, our "rise over run" is 31​. We can then substitute those pieces of information into our formula and we get: y=31​x+1.
Exercises 3 An equation in slope-intercept form follows a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=2 and b=9 into this equation, we can write the line's slope-intercept form equation. y=2x+9​
Exercises 4 Equations in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=0 and b=5 into this equation, we can write the line's slope-intercept form equation. y=0x+5 ⇒y=5​
Exercises 5 An equation in slope-intercept form follows a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=-3 and b=0 into this equation, we can write the line's slope-intercept form equation. y=-3x+0 ⇒y=-3x​
Exercises 6 Equations in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=-7 and b=1 into this equation, we can write the line's slope-intercept form equation. y=-7x+1​
Exercises 7 Equations in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=32​ and b=-8 into this equation, we can write the line's slope-intercept form equation. y=32​x+(-8) ⇒y=32​x−8​
Exercises 8 Equations in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope and b is the y-intercept. By substituting m=-43​ and b=-6 into this equation, we can write the line's slope-intercept form equation. y=-43​x+(-6) ⇒y=-43​x−6​
Exercises 9 Let's start by recalling the slope-intercept form of a line. y=mx+b​ Here, m is the slope and b the y-intercept. Let's find these two values for the given line.Finding the y-intercept Consider the given graph.We can see that the line intercepts the y-axis at (0,2). This means that b=2. y=mx+2​Finding the Slope To find the slope, we will use the fact that the line also passes through the point (3,3). We will be able to identify the slope m using the rise and run of the graph.Traveling from the y-intercept to the point (3,3) requires moving 3 steps to the right and 1 step up. runrise​=31​⇔m=31​​ We can now write the complete equation of the line. y=31​x+2​
Exercises 10 Equations written in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope of the line and b is the y-intercept. We need to identify these values using the graph. Let's start with the y-intercept.Finding the y-intercept Observe the given graph.We can see that the function intercepts the y-axis at (0,3). This means that the value of b is 3.Finding the Slope To find the slope, we will trace along the line on the given graph until we find a lattice point, which is a point that lies perfectly on the grid lines. In doing so, we will be able to identify the slope m using the rise and run of the graph.Here we've identified (4,2) as our other point. Traveling to this point from the y-intercept requires that we move 4 steps horizontally in the positive direction and 1 step vertically in the negative direction. runrise​=4-1​⇔m=-41​​Writing the Equation Now that we have the slope and the y-intercept, we can form our final equation. y=mx+by=-41​x+3​
Exercises 11 Let's start by recalling the slope-intercept form of a line. y=mx+b​ Here, m is the slope and b the y-intercept. Let's find these two values for the given line.Finding the y-intercept Consider the given graph.We can see that the line intercepts the y-axis at (0,0). This means that b=0. y=mx+0⇔y=mx​Finding the Slope To find the slope, we will use the fact that the line also passes through the point (-3,4). We will be able to identify the slope m using the rise and run of the graph.Traveling from the y-intercept to the point (-3,4) requires moving 3 steps to the left and 4 steps up. runrise​=-34​⇔m=-34​​ We can now write the complete equation of the line. y=-34​x​
Exercises 12 Equations written in slope-intercept form follow a specific format. y=mx+b​ In this form, m is the slope of the line and b is the y-intercept. We need to identify these values using the graph. Let's start with the y-intercept.Finding the y-intercept Observe the given graph.We can see that the function intercepts the y-axis at (0,-2). This means that the value of b is -2.Finding the Slope To find the slope, we will trace along the line on the given graph until we find a lattice point, which is a point that lies perfectly on the grid lines. In doing so, we will be able to identify the slope m using the rise and run of the graph.Here we've identified (2,2) as our other point. Traveling to this point from the y-intercept requires that we move 2 steps horizontally in the positive direction and 4 steps vertically in the positive direction. runrise​=24​⇔m=2​Writing the Equation Now that we have the slope and the y-intercept, we can form our final equation. y=mx+by=2x+(-2) ⇒y=2x−2​
Exercises 13 An equation in slope-intercept form follows a specific format. y=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (3,1) & (0,10)m=0−310−1​ Simplify RHS Subtract termsm=-39​Calculate quotient m=-3 A slope of -3 means that for every 1 horizontal step in the positive direction, we take 3 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. y=-3x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,10), we already know that we have a y-intercept of 10. We can now complete the equation. y=-3x+10​
Exercises 14 An equation in slope-intercept form follows a specific format. y=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (2,7) & (0,-5)m=0−2-5−7​ Simplify RHS Subtract termsm=-2-12​Calculate quotient m=6 A slope of 6 means that for every 1 horizontal step in the positive direction, we take 6 vertical steps in the positive direction. Now that we know the slope, we can write a partial version of the equation. y=6x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,-5), we already know that we have a y-intercept of -5. We can now complete the equation. y=6x+(-5) ⇒y=6x−5​
Exercises 15 The equation of a line written in slope-intercept form follows a specific format. y=mx+b​ In the above formula, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula to calculate the slope m. m=x2​−x1​y2​−y1​​Substitute (2,-4) & (0,-4)m=0−2-4−(-4)​ Simplify right-hand side a−(-b)=a+bm=0−2-4+4​Add and subtract termsm=-20​Calculate quotient m=0 A slope of 0 means that for every 1 horizontal step, we take 0 vertical steps. Therefore, we have a horizontal line. Now that we know the slope, we can partially write the equation of the line. y=0x+b​ To complete the equation, we also need to determine the value of the y-intercept, b. One of the given points is (0,-4). Therefore, the y-intercept is b = -4. We can now complete the equation. y=0x+(-4)⇔y=-4​
Exercises 16 An equation in slope-intercept form follows a specific format. y=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (-6,0) & (0,-24)m=0−(-6)-24−0​ Simplify RHS a−(-b)=a+bm=0+6-24−0​Add and subtract termsm=6-24​Calculate quotient m=-4 A slope of -4 means that for every 1 horizontal step in the positive direction, we take 4 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. y=-4x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,-24), we already know that we have a y-intercept of -24. We can now complete the equation. y=-4x+(-24) ⇒y=-4x−24​
Exercises 17 An equation in slope-intercept form follows a specific format. y=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,5) & (-1.5,1)m=-1.5−01−5​ Simplify RHS Subtract termsm=-1.5-4​ba​=b/-0.5a/-0.5​ m=38​ A slope of 38​ means that for every 3 horizontal steps in the positive direction, we take 8 vertical steps in the positive direction. Now that we know the slope, we can write a partial version of the equation. y=38​x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,5), we already know that we have a y-intercept of 5. We can now complete the equation. y=38​x+5​
Exercises 18 An equation in slope-intercept form follows a specific format. y=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,3) & (-5,2.5)m=-5−02.5−3​ Simplify RHS Subtract termsm=-5-0.5​Calculate quotientm=0.1Write as a fraction m=101​ A slope of 101​ means that for every 10 horizontal steps in the positive direction, we take 1 vertical step in the positive direction. Now that we know the slope, we can write a partial version of the equation. y=101​x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,3), we already know that we have a y-intercept of 3. We can now complete the equation. y=101​x+3​
Exercises 19 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(0)=2⇔(0,2)f(2)=4⇔(2,4)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,2) & (2,4)m=2−04−2​ Simplify RHS Subtract termsm=22​Calculate quotient m=1 A slope of 1 means that for every 1 horizontal step in the positive direction, we take 1 vertical step in the positive direction. Now that we know the slope, we can write a partial version of the equation. f(x)=1x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,2), we already know that we have a y-intercept of 2. We can now complete the equation. f(x)=1x+2 ⇒f(x)=x+2​
Exercises 20 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(0)=7⇔(0,7)f(3)=1⇔(3,1)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,7) & (3,1)m=3−01−7​ Simplify RHS Subtract termsm=3-6​Calculate quotient m=-2 A slope of -2 means that for every 1 horizontal step in the positive direction, we take 2 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. f(x)=-2x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,7), we already know that we have a y-intercept of 7. We can now complete the equation. f(x)=-2x+7​
Exercises 21 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(4)=-3⇔(4,-3)f(0)=-2⇔(0,-2)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (4,-3) & (0,-2)m=0−4-2−(-3)​ Simplify RHS a−(-b)=a+bm=0−4-2+3​Add and subtract termsm=-41​Put minus sign in front of fraction m=-41​ A slope of -41​ means that for every 4 horizontal steps in the positive direction, we take 1 vertical step in the negative direction. Now that we know the slope, we can write a partial version of the equation. f(x)=-41​x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,-2), we already know that we have a y-intercept of -2. We can now complete the equation. f(x)=-41​x+(-2) ⇒f(x)=-41​x−2​
Exercises 22 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(5)=-1⇔(5,-1)f(0)=-5⇔(0,-5)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (5,-1) & (0,-5)m=0−5-5−(-1)​ Simplify RHS a−(-b)=a+bm=0−5-5+1​Add and subtract termsm=-5-4​-b-a​=ba​ m=54​ A slope of 54​ means that for every 5 horizontal steps in the positive direction, we take 4 vertical steps in the positive direction. Now that we know the slope, we can write a partial version of the equation. f(x)=54​x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,-5), we already know that we have a y-intercept of -5. We can now complete the equation. f(x)=54​x+(-5) ⇒f(x)=54​x−5​
Exercises 23 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(-2)=6⇔(-2,6)f(0)=-4⇔(0,-4)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (-2,6) & (0,-4)m=0−(-2)-4−6​ Simplify RHS a−(-b)=a+bm=0+2-4−6​Add and subtract termsm=2-10​Calculate quotient m=-5 A slope of -5 means that for every 1 horizontal step in the positive direction, we take 5 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. f(x)=-5x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,-4), we already know that we have a y-intercept of -4. We can now complete the equation. f(x)=-5x+(-4) ⇒f(x)=-5x−4​
Exercises 24 We are asked to write an equation using function notation. Let's recall the slope-intercept form of a line. f(x)=mx+b​ In the above formula, m represents the slope and b the y-intercept of the line. We have been given two points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(0)=3⇔(0,3)f(-6)=3⇔(-6,3)​ Let's use the given points to calculate m and b. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,3) & (-6,3)m=-6−03−3​Subtract termsm=-60​Calculate quotientm=0 A slope of 0 means that we have a horizontal line. f(x)=0x+b​ To complete the equation, we also need to determine the y-intercept b. Since one of the given points is (0,3), we know that the y-intercept is 3. f(x)=0x+3⇔f(x)=3​
Exercises 25 The easiest way to write a linear function is in the slope-intercept form, y=mx+b. The slope m is the value expressing the RunRise​=Change in xChange in y​. In the given mapping, we can see that for each increase by 1 unit for x, the values for y decrease by 2: x-values: y-values: ​-1→+10→+11.3→−21→−2-1​ This rate of change means that our function has a slope of -2. We can also see this using the formula for slope and two of the points. m=x2​−x1​y2​−y1​​Substitute (1,-1) & (0,1)m=1−0-1−1​Subtract termsm=1-2​1a​=am=-2 We now have that our equation is: f(x)=-2x+b. The y-intercept b is the point where x=0, which we have been given in the mapping, (0,1). We can substitute this value into the slope-intercept form and have a final equation of: f(x)=-2x+1.
Exercises 26 In this case, we are asked to write our equation in function notation. f(x)=mx+b​ For an equation in this form, m is the slope and b is the y-intercept. We have also been given three points in function notation. To write these points as coordinate pairs, remember that the input x is the x-coordinate and the output f(x) is the y-coordinate. f(x)=y⇔(x,y)f(-4)=-2⇔(-4,-2)f(-2)=-1⇔(-2,-1)f(0)=0⇔(0,0)​ Let's use the first two given points to calculate m. We will start by substituting the points into the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (-4,-2) & (-2,-1)m=-2−(-4)-1−(-2)​a−(-b)=a+bm=-2+4-1+2​Add termsm=21​ A slope of 21​ means that for every 2 horizontal steps in the positive direction, we take 1 vertical step in the positive direction. Now that we know the slope, we can write a partial version of the equation. f(x)=21​x+b​ To complete the equation, we also need to determine the y-intercept, b. Since we know that one of the given points is (0,0) we already know that we have a y-intercept of 0. We can now complete the equation. f(x)=21​x+(0) ⇒f(x)=21​x​
Exercises 27 To describe and correct the error, we will write the equation of the line and then compare it with the given one. Let's start by recalling the slope-intercept form of a line. y=mx+b​ Here, m is the slope and b the y-intercept. We are told the the slope is 2 and the y-intercept is 7. Therefore, in the above formula, we can substitute 2 and 7 for m and b, respectively. y=2x+7​ The error was substituting the slope and y-intercept incorrectly in the slope-intercept form.
Exercises 28 We find the slope of a line, or a line segment, by dividing the vertical change by the horizontal change between two points on the line. We can use the Slope Formula for this purpose: Slope=x2​−x1​y2​−y1​​ The two points on the line are given as (0,4) and (5,1). When we substitute them into the above formula we get: Slope=5−01−4​=-53​ It's a negative slope, as it should be, because the line is sloping downward. The mistake made in the shown solution was in calculating the slope. The points did not line up accordingly. The coordinates must be put into the Slope Formula in the correct order.
Exercises 29 We are asked to write a linear model that represents the world record (in minutes) for the men's mile as a function of the number of years since 1960. Then we are asked to use it to estimate the record time in 2000 and to predict the record time in 2020.Writing the linear model To write the linear model, we will use the slope-intercept form of a line, where m is the slope and b is the y-intercept. y=mx+b Let x represent the time in years since 1960 and let y represent the record for the men's mile in minutes. Since x is defined in years since 1960, 1960 corresponds to x=0, when the men's record was 3.91 minutes. This means we have the point (0,3.91) and, therefore, the y-intercept is 3.91. So far we have y=mx+3.91. In 1980, the record was 3.81 minutes. As 1980 is 20 years after 1960, it corresponds to x=20. This means we have the point (20,3.81). We can find the slope using both points and the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,3.91) & (20,3.81)m=20−03.81−3.91​ Simplify RHS Subtract termsm=20-0.1​Use a calculator m=-0.005 Therefore, the linear model can be written as follows. y=-0.005x+3.91Using the linear model We will use our linear model to estimate the men's world record in the year 2000 and to predict the record for the year 2020.Year 2000 Since 2000 is 40 years after 1960, it corresponds to x=40. As a consequence, we will substitute 40 for x in the linear model to find the estimation. y=-0.005x+3.91x=40y=-0.005⋅40+3.91(-a)b=-aby=-0.2+3.91Add termsy=3.71 Therefore, the estimated men's world record of the year 2000 is 3.71 minutes.Year 2020 Since 2020 is 60 years after 1960, it corresponds to x=60. As a consequence, we will substitute 60 for x in the linear model to find the prediction. y=-0.005x+3.91x=60y=-0.005⋅60+3.91(-a)b=-aby=-0.3+3.91Add termsy=3.61 Therefore, the prediction men's world record for the year 2020 is 3.61 minutes.
Exercises 30
Exercises 31 Let's calculate the slope first. If we are given two points on a line, we can use this formula to find the slope: Slope=x2​−x1​y2​−y1​​ The two points on the line are given as (0,-2) and (0,5). Let's substitute these into the Slope Formula. Slope=x2​−x1​y2​−y1​​Substitute (0,-2) & (0,5)Slope=0−0-2−5​Subtract termsSlope=0-7​ Because dividing by 0 is not allowed (it's a big "no no"), the slope is undefined. The line through the given points is vertical. Its equation is x=0, which is not in slope-intercept form.
Exercises 32 Let's pretend that x is the number of nights we are booking a hotel and y is the total cost. Since the graph passes through the point (0,20), we can assume that there is a booking or processing fee of $20. Now, let's calculate the slope of this line using the slope-formula. m=x2​−x1​y2​−y1​​Substitute (0,20) & (4,80)m=4−080−20​Subtract termsm=460​Calculate quotientm=15 The slope is 15. This means that booking the hotel for each additional night will increase the total bill by $15.
Exercises 33 We want to write the slope-intercept form for Ax+By=C. Then, we want to use this answer to find the slope and y-intercept of the graph of the equation -6x+5y=9.Writing Ax+By=C in Slope-intercept Form To write Ax+By=C in slope-intercept form, we have to isolate the y-variable. To do so, we need to assume that B​=0, since division by zero is not defined. Ax+By=C Write in slope-intercept form LHS−Ax=RHS−AxBy=-Ax+CLHS/B=RHS/By=B-Ax+C​Write as a sum of fractionsy=B-A​x+BC​Put minus sign in front of fraction y=-BA​x+BC​ We found the slope-intercept form of the given line. Standard form:Slope-intercept form:​ Ax+By=C y=-BA​x+BC​,   B​=0​ In the second of the above equations, the slope of the line is -BA​ and the y-intercept is BC​.Finding the Slope and y-intercept of -6x+5y=9 Note that the given equation is written in standard form. Let's identify the values of A, B, and C. -6x+5y=9​ We see that A=-6, B=5, and C=9. By substituting these values into the expressions -BA​ and BC​, we can determine the slope and the y-intercept of the line. Let's start with the slope. slope=-BA​A=-6, B=5slope=-5-6​-b-a​=ba​slope=56​ The slope of the line is 56​. Let's now substitute 9 for C and 5 for B in BC​ to obtain the y-intercept. y-intercept=BC​C=9, B=5y-intercept=59​ The y-intercept of the line is 59​.
Exercises 34 Let's unpack their ideas on at a time.Our friend Having two points on a linear function, we can write its equation in a slope-intercept form by determining the slope using the Slope Formula. Then we can substitute one of the points into the equation and isolate the y-intercept. Our friend is correct because they have qualified the line as being a linear function. A linear function can only have one output for each input.Our Cousin Our cousin claims that the two points could lie on a vertical line. Let's graph an example of this situation.On the above graph, x=2, we have infinitely many outputs. Two outputs have been marked on the graph. However, this is not a linear function. Our cousin is incorrect because they are forgetting that the line is already stated to be a function.
Exercises 35 Let's start by reflecting line ℓ in the x-axis. This will give us the graph of line k. Note that symmetric points will have the same x-coordinate, but their y-coordinates will have opposite signs.To find the equation of line k, we need to find its slope and y-intercept. We see above that line k intercepts the y-axis at the point (0,-1). Therefore, the y-intercept is -1. y=mx+(-1)⇔y=mx−1​ To find the slope m, we will substitute two of its points, (0,-1) and (3,4), in the Slope Formula. m=x2​−x1​y2​−y1​​Substitute (0,-1) & (3,4)m=3−04−(-1)​a−(-b)=a+bm=3−04+1​Add and subtract termsm=35​ Now that we know the slope is m=35​, we can write the equation of line k. y=35​x−1​
Exercises 36
Exercises 37 We have been given the points (0,b) and (1,b+m) and asked to show that the equation of this line is y=mx+b. The slope-intercept form of a line can be thought of as: y=Slope⋅x+the y-intercept. The y-intercept of any line is the point at which the line crosses the y-axis. This is always when x=0. Therefore, having been given that the point (0,b) lies on the line, we know that when x=0, y=b and our y-intercept is b. Our equation then becomes: y=Slope⋅x+b. We can use the Slope Formula and the given points to find the slope. m=x2​−x1​y2​−y1​​Substitute (0,b) & (1,b+m)m=1−0b+m−b​Add and subtract termsm=1m​1a​=am=m The entire equation is then: y=mx+b.How can we be sure about (-1,b−m)? Let's substitute the point (-1,b−m) into the equation so that we can test whether the statement holds true. y=mx+bx=-1, y=b−mb−m=?m(-1)+b(-a)b=-abb−m=?-m+bCommutative Property of Additionb−m=b−m We have reached an identity statement, b−m will always equal b−m. Therefore, we know that the point lies on the line no matter the values for b and m.
Exercises 38 To solve an equation, we need to isolate the variable. In the given equation, notice that there are two variable terms. We will need to move them both to the same side. We will start by distributing 3. 3(x−15)=x+11Distribute 33x−45=x+11LHS−x=RHS−x3x−45−x=x+11−xSubtract term2x−45=11LHS+45=RHS+452x−45+45=11+45Add terms2x=56LHS/2=RHS/222x​=256​Calculate quotientx=28 The solution to the equation is x=28.
Exercises 39 To solve an equation, we need to isolate the variable. In the given equation, notice that there are two variable terms. We will need to move them both to the same side. We will start by distributing 4. -4y−10=4(y−3)Distribute 4-4y−10=4y−12LHS+4y=RHS+4y-4y−10+4y=4y−12+4yAdd terms-10=8y−12LHS+12=RHS+12-10+12=8y−12+12Add terms2=8yLHS/8=RHS/882​=88y​Calculate quotient41​=yRearrange equationy=41​ The solution to the equation is y=41​.
Exercises 40 To solve an equation, we need to isolate the variable. In the given equation, notice that there are two variable terms. We will need to move them both to the same side. We will start by distributing 2. 2(3d+3)=7+6dDistribute 26d+6=7+6dLHS−6d=RHS−6d6d+6−6d=7+6d−6dSubtract term6​=7 The equation simplified to a false statement. This means that there is no solution to this equation.
Exercises 41 To solve an equation, we need to isolate the variable. In the given equation, notice that there are two variable terms. We will need to move them both to the same side. We will start by distributing -5 and 10. -5(4−3n)=10(n−2)Distribute -5-20+15n=10(n−2)Distribute 10-20+15n=10n−20LHS−10n=RHS−10n-20+15n−10n=10n−20−10nSubtract term-20+5n=-20LHS+20=RHS+20-20+5n+20=-20+20Add terms5n=0LHS/5=RHS/555n​=50​Calculate quotientn=0 The solution to the equation is n=0.
Exercises 42 We will graph this equation by finding and plotting its intercepts, then connecting them with a line. To find the x- and y-intercepts, we will need to substitute 0 for one variable, solve, then repeat for the other variable.Finding the x-intercept Think of the point where the graph of an equation crosses the x-axis. The y-value of that (x,y) coordinate pair is equal to 0, and the x-value is the x-intercept. To find the x-intercept of the given equation, we should substitute 0 for y and solve for x. -4x+2y=16y=0-4x+2(0)=16Zero Property of Multiplication-4x=16LHS/-4=RHS/-4x=-4 An x-intercept of -4 means that the graph passes through the x-axis at the point (-4,0).Finding the y-intercept Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the (x,y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept. -4x+2y=16x=0-4(0)+2y=16Zero Property of Multiplication2y=16LHS/2=RHS/2y=8 A y-intercept of 8 means that the graph passes through the y-axis at the point (0,8).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 43 We will graph this equation by finding and plotting its intercepts, then connecting them with a line. To find the x- and y-intercepts, we will need to substitute 0 for one variable, solve, then repeat for the other variable.Finding the x-intercept Think of the point where the graph of an equation crosses the x-axis. The y-value of that (x,y) coordinate pair is equal to 0, and the x-value is the x-intercept. To find the x-intercept of the given equation, we should substitute 0 for y and solve for x. 3x+5y=-15y=03x+5(0)=-15Zero Property of Multiplication3x=-15LHS/3=RHS/3x=-5 An x-intercept of -5 means that the graph passes through the x-axis at the point (-5,0).Finding the y-intercept Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the (x,y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept. 3x+5y=-15x=03(0)+5y=-15Zero Property of Multiplication5y=-15LHS/5=RHS/5y=-3 A y-intercept of -3 means that the graph passes through the y-axis at the point (0,-3).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 44 We will graph this equation by finding and plotting its intercepts, then connecting them with a line. To find the x- and y-intercepts, we will need to substitute 0 for one variable, solve, then repeat for the other variable.Finding the x-intercept Think of the point where the graph of an equation crosses the x-axis. The y-value of that (x,y) coordinate pair is equal to 0, and the x-value is the x-intercept. To find the x-intercept of the given equation, we should substitute 0 for y and solve for x. x−6y=24y=0x−6(0)=24Zero Property of Multiplicationx=24 An x-intercept of 24 means that the graph passes through the x-axis at the point (24,0).Finding the y-intercept Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the (x,y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept. x−6y=24x=00−6y=24Zero Property of Multiplication-6y=24LHS/-6=RHS/-6y=-4 A y-intercept of -4 means that the graph passes through the y-axis at the point (0,-4).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 45 We will graph this equation by finding and plotting its intercepts, then connecting them with a line. To find the x- and y-intercepts, we will need to substitute 0 for one variable, solve, then repeat for the other variable.Finding the x-intercept Think of the point where the graph of an equation crosses the x-axis. The y-value of that (x,y) coordinate pair is equal to 0, and the x-value is the x-intercept. To find the x-intercept of the given equation, we should substitute 0 for y and solve for x. -7x−2y=-21y=0-7x−2(0)=-21Zero Property of Multiplication-7x=-21LHS/-7=RHS/-7x=3 An x-intercept of 3 means that the graph passes through the x-axis at the point (3,0).Finding the y-intercept Let's use the same concept to find the y-intercept. Consider the point where the graph of the equation crosses the y-axis. The x-value of the (x,y) coordinate pair at the y-intercept is 0. Therefore, substituting 0 for x will give us the y-intercept. -7x−2y=-21x=0-7(0)−2y=-21Zero Property of Multiplication-2y=-21LHS/-2=RHS/-2y=221​ A y-intercept of 221​ means that the graph passes through the y-axis at the point (0,221​).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
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