#### Graphing Linear Equations in Standard Form

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Exercises 1 Let's first talk about each intercept a bit.x-intercept An x-intercept of a function is the point at which the function crosses the x-axis. It is always (x,0) for some value of x. It tells us when the input of a function is 0.y-intercept A y-intercept of a function is the point at which the function crosses the y-axis. It is always (0,y) for some value of y. It tells us when the output of a function is 0. With that said we can identify the following similarities and differences:Alike: They are both the points where an axis is crossed. Different: They cross different axes. One tells input and one tells output.
Exercises 2 Let's plot the given points on a coordinate plane so that we can visualize which one doesn't belong.These points form a perfect square and, therefore, each point is a combination of numbers from two other points. The biggest difference is that the point (4,-3) does not fall on an axis. It's the one that doesn't belong.
Exercises 3 In this exercise, we are asked to graph an equation with only one variable term. This means that the graph will either be horizontal or vertical. To graph it, let's first rewrite the given equation in standard form, Ax+By=C. x=4⇔1x+0y=4​ Let's use this equation and substitute some values for y. Doing so will generate some coordinate pairs for us to use in our graph.y1x+0y=4Simplify(x,y) 01x+0(0)=4x=4(4,0) 21x+0(2)=4x=4(4,2) 41x+0(4)=4x=4(4,4) 61x+0(6)=4x=4(4,6) Now we will plot these points and connect them with a line to form our graph.
Exercises 4 In this exercise, we are asked to graph an equation with only one variable term. This means that the graph will either be horizontal or vertical. To graph it, let's first rewrite the given equation in standard form, Ax+By=C. y=2⇔0x+1y=2​ Let's use this equation and substitute a few arbitrary values for x. Doing so will generate some coordinate pairs for us to use in our graph.x0x+1y=2Simplify(x,y) 00(0)+1y=2y=2(0,2) 20(2)+1y=2y=2(2,2) 40(4)+1y=2y=2(4,2) 60(6)+1y=2y=2(6,2) Now we will plot these points and connect them with a line to form our graph.
Exercises 5 In this exercise, we are asked to graph an equation with only one variable term. This means that the graph will either be horizontal or vertical. To graph it, let's first rewrite the given equation in standard form, Ax+By=C. y=-3⇔0x+1y=-3​ Let's use this equation and substitute a few arbitrary values for x. Doing so will generate some coordinate pairs for us to use in our graph.x0x+1y=-3Simplify(x,y) 00(0)+1y=-3y=-3(0,-3) 20(2)+1y=-3y=-3(2,-3) 40(4)+1y=-3y=-3(4,-3) 60(6)+1y=-3y=-3(6,-3) Now we will plot these points and connect them with a line to form our graph.
Exercises 6 In this exercise, we are asked to graph an equation with only one variable term. This means that the graph will either be horizontal or vertical. To graph it, let's first rewrite the given equation in standard form, Ax+By=C. x=-1⇔1x+0y=-1​ Let's use this equation and substitute some values for y. Doing so will generate some coordinate pairs for us to use in our graph.y1x+0y=-1Simplify(x,y) 01x+0(0)=-1x=-1(-1,0) 21x+0(2)=-1x=-1(-1,2) 41x+0(4)=-1x=-1(-1,4) 61x+0(6)=-1x=-1(-1,6) Now we will plot these points and connect them with a line to form our graph.
Exercises 7 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. 2x+3y=12y=02x+3(0)=12Zero Property of Multiplication2x=12LHS/2=RHS/2x=6 An x-intercept of 6 means that the graph passes through the x-axis at the point (6,0).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. 2x+3y=12x=02(0)+3y=12Zero Property of Multiplication3y=12LHS/3=RHS/3y=4 A y-intercept of 4 means that the graph passes through the y-axis at the point (0,4).
Exercises 8 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. 3x+6y=24y=03x+6(0)=24Zero Property of Multiplication3x=24LHS/3=RHS/3x=8 An x-intercept of 8 means that the graph passes through the x-axis at the point (8,0).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+6y=24x=03(0)+6y=24Zero Property of Multiplication6y=24LHS/6=RHS/6y=4 A y-intercept of 4 means that the graph passes through the y-axis at the point (0,4).
Exercises 9 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. -4x+8y=-16y=0-4x+8(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+8y=-16x=0-4(0)+8y=-16Zero Property of Multiplication8y=-16LHS/8=RHS/8y=-2 A y-intercept of -2 means that the graph passes through the y-axis at the point (0,-2).
Exercises 10 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. -6x+9y=-18y=0-6x+9(0)=-18Zero Property of Multiplication-6x=-18LHS/-6=RHS/-6x=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. -6x+9y=-18x=0-6(0)+9y=-18Zero Property of Multiplication9y=-18LHS/9=RHS/9y=-2 A y-intercept of -2 means that the graph passes through the y-axis at the point (0,-2).
Exercises 11 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. 3x−6y=2y=03x−6(0)=2Zero Property of Multiplication3x=2LHS/3=RHS/3x=32​ An x-intercept of 32​ means that the graph passes through the x-axis at the point (32​,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−6y=2x=03(0)−6y=2Zero Property of Multiplication-6y=2LHS/-6=RHS/-6y=-62​Put minus sign in front of fractiony=-62​ba​=b/2a/2​y=-31​ A y-intercept of -31​ means that the graph passes through the y-axis at the point (0,-31​).
Exercises 12 To determine the x- and y-intercepts for the equation, 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 0, and the x-value is the x-intercept. To find the x-intercept of the equation, we should substitute 0 for y and solve for x. -x+8y=4y=0-x+8(0)=4Zero Property of Multiplication-x=4Change signsx=-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. -x+8y=4x=0-(0)+8y=4Zero Property of Multiplication8y=4LHS/8=RHS/8y=84​ba​=b/4a/4​y=21​ A y-intercept of 21​ means that the graph passes through the y-axis at the point (0,21​).
Exercises 13 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. 5x+3y=30y=05x+3(0)=30Zero Property of Multiplication5x=30LHS/5=RHS/5x=6 An x-intercept of 6 means that the graph passes through the x-axis at the point (6,0).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. 5x+3y=30x=05(0)+3y=30Zero Property of Multiplication3y=30LHS/3=RHS/3y=10 A y-intercept of 10 means that the graph passes through the y-axis at the point (0,10).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 14 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+6y=12y=04x+6(0)=12Zero Property of Multiplication4x=12LHS/4=RHS/4x=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. 4x+6y=12x=04(0)+6y=12Zero Property of Multiplication6y=12LHS/6=RHS/6y=2 A y-intercept of 2 means that the graph passes through the y-axis at the point (0,2).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 15 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. -12x+3y=24y=0-12x+3(0)=24Zero Property of Multiplication-12x=24LHS/-12=RHS/-12x=-2 An x-intercept of -2 means that the graph passes through the x-axis at the point (-2,0).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. -12x+3y=24x=0-12(0)+3y=24Zero Property of Multiplication3y=24LHS/3=RHS/3y=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 16 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. -2x+6y=18y=0-2x+6(0)=18Zero Property of Multiplication-2x=18LHS/-2=RHS/-2x=-9 An x-intercept of -9 means that the graph passes through the x-axis at the point (-9,0).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. -2x+6y=18x=0-2(0)+6y=18Zero Property of Multiplication6y=18LHS/6=RHS/6y=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 17 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+3y=-30y=0-4x+3(0)=-30Zero Property of Multiplication-4x=-30LHS/-4=RHS/-4x=430​ba​=b/2a/2​x=215​ An x-intercept of 215​ means that the graph passes through the x-axis at the point (215​,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+3y=-30x=0-4(0)+3y=-30Zero Property of Multiplication3y=-30LHS/3=RHS/3y=-10 A y-intercept of -10 means that the graph passes through the y-axis at the point (0,-10).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 18 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. -2x+7y=-21y=0-2x+7(0)=-21Zero Property of Multiplication-2x=-21LHS/-2=RHS/-2x=221​ An x-intercept of 221​ means that the graph passes through the x-axis at the point (221​,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. -2x+7y=-21x=0-2(0)+7y=-21Zero Property of Multiplication7y=-21LHS/7=RHS/7y=-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 19 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+2y=7y=0-x+2(0)=7Zero Property of Multiplication-x=7Change signsx=-7 An x-intercept of -7 means that the graph passes through the x-axis at the point (-7,0).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+2y=7x=0-(0)+2y=7Zero Property of Multiplication2y=7LHS/2=RHS/2y=27​ A y-intercept of 27​ means that the graph passes through the y-axis at the point (0,27​).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 20 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−y=-5y=03x−(0)=-5Zero Property of Multiplication3x=-5LHS/3=RHS/3x=3-5​Put minus sign in front of fractionx=-35​ An x-intercept of -35​ means that the graph passes through the x-axis at the point (-35​,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−y=-5x=03(0)−y=-5Zero Property of Multiplication-y=-5Change signsy=5 A y-intercept of 5 means that the graph passes through the y-axis at the point (0,5).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 21 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. -25​x+y=10y=0-25​x+0=10Zero Property of Multiplication-25​x=10LHS⋅2=RHS⋅2-5x=20LHS/-5=RHS/-5x=-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. -25​x+y=10x=0-25​(0)+y=10Zero Property of Multiplicationy=10 A y-intercept of 10 means that the graph passes through the y-axis at the point (0,10).Graphing the equation We can now graph the equation by plotting the intercepts and connecting them with a line.
Exercises 22 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. -21​x+y=-4y=0-21​x+0=-4Zero Property of Multiplication-21​x=-4LHS⋅2=RHS⋅2-x=-8Change signsx=8 An x-intercept of 8 means that the graph passes through the x-axis at the point (8,0).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. -21​x+y=-4x=0-21​(0)+y=-4Zero Property of Multiplicationy=-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 23
Exercises 24
Exercises 25 The given equation, 3x+12y=24. represents a straight line on the plane and it has two intercepts. In the given solution, both axes' intercepts were calculated in a correct manner but they were not written properly. With any non-horizontal linear function, the x-intercept has y=0 and the y-intercept has x=0. You don't combine the intercepts to form an "intercept-point." Therefore, the correct answer is: ​x-intercept: (8,0)y-intercept: (0,2).​
Exercises 26 Both axes' intercepts are calculated correctly but were not written properly. The points in plane geometry are written as an ordered pair. The first number is the x-coordinate and the second one is the y-coordinate. Therefore, if we find x to be 5 when y is 0, then the point should be written as (5,0), not (0,5). Similarly, the y-intercept should be written as (0,2), not (2,0).
Exercises 27 The intercepts of a function tell us where the graph will cross each axis. The x-intercept tells us where the the graph will cross the x-axis. This is when y=0. Therefore, when you want to find the x-intercept we have to substitute 0 for y and continue solving the equation for x. Our friend had it backwards.
Exercises 28
Exercises 29 Each graph has a different set of x- and y-intercepts. By finding these, we can identify which graph belongs to which equation.x-intercept The x-intercept occurs when y=0. By substituting 0 for y into our equation and solving for x, we can find the x-intercept. 5x+3y=30y=05x+3(0)=30 Solve for x Zero Property of Multiplication5x+0=30Identity Property of Addition5x=30LHS/5=RHS/5 x=6 The x-intercept is at the point (6,0).y-intercept To find the y-intercept, we substitute 0 for x and solve for y. 5x+3y=30x=05(0)+3y=30 Solve for y Zero Property of Multiplication0+3y=30Identity Property of Addition3y=30LHS/3=RHS/3 y=10 The y-intercept is at the point (0,10).Graph Now that we know the intercepts, we can draw the graph of this equation.This graph corresponds to option A.
Exercises 30 Each graph has a different set of x- and y-intercepts. By finding these, we can identify which graph belongs to which equation.x-intercept The x-intercept occurs when y=0. By substituting 0 for y into our equation and solving for x, we can find the x-intercept. 5x+3y=-30y=05x+3(0)=-30 Solve for x Zero Property of Multiplication5x+0=-30Identity Property of Addition5x=-30LHS/5=RHS/5 x=-6 The x-intercept is at the point (-6,0).y-intercept To find the y-intercept, we substitute 0 for x and solve for y. 5x+3y=-30x=05(0)+3y=-30 Solve for y Zero Property of Multiplication0+3y=-30Identity Property of Addition3y=-30LHS/3=RHS/3 y=-10 The y-intercept is at the point (0,-10).Graph Now that we know the intercepts, we can draw the graph of this equation.This graph corresponds to option C.
Exercises 31 Each graph has a different set of x- and y-intercepts. By finding these, we can identify which graph belongs to which equation.x-intercept The x-intercept occurs when y=0. By substituting 0 for y into our equation and solving for x, we can find the x-intercept. 5x−3y=30y=05x−3(0)=30 Solve for x Zero Property of Multiplication5x−0=30Subtract term5x=30LHS/5=RHS/5 x=6 The x-intercept is at the point (6,0).y-intercept To find the y-intercept, we substitute 0 for x and solve for y. 5x−3y=30x=05(0)−3y=30 Solve for y Zero Property of Multiplication0−3y=30Subtract term-3y=30LHS/(-3)=RHS/(-3) y=-10 The y-intercept is at the point (0,-10).Graph Now that we know the intercepts, we can draw the graph of this equation.This graph corresponds to option D.
Exercises 32 Each graph has a different set of x- and y-intercepts. By finding these, we can identify which graph belongs to which equation.x-intercept The x-intercept occurs when y=0. By substituting 0 for y into our equation and solving for x, we can find the x-intercept. 5x−3y=-30y=05x−3(0)=-30 Solve for x Zero Property of Multiplication5x−0=-30Subtract term5x=-30LHS/5=RHS/5 x=-6 The x-intercept is at the point (-6,0).y-intercept To find the y-intercept, we substitute 0 for x and solve for y. 5x−3y=-30x=05(0)−3y=-30 Solve for y Zero Property of Multiplication0−3y=-30Subtract term-3y=-30LHS/(-3)=RHS/(-3) y=10 The y-intercept is at the point (0,10).Graph Now that we know the intercepts, we can draw the graph of this equation.This graph corresponds to option B.
Exercises 33 Horizontal lines have equations in the form: y=k, where k is any real number. Therefore, the given equations y=-2 and y=1 are represented by horizontal lines in the coordinate plane.Now we can graph the other two equations. Vertical lines have equations in the form x=k, where k is any real number. Therefore, the given equations x=5 and x=2 are represented by vertical lines in the coordinate plane.As we can see, the horizontal and vertical lines form a rectangle. Additionally, since the distance from x=2 and x=5 is the same as the distance between y=-2 and y=1, the rectangle is in fact a square.
Exercises 34
Exercises 35 The given equation represents a straight line and it crosses both axes at some point. We were given the following equation: 22​ x+22​ y=30. Let's find its intercepts one at a time.x-intercept In the exercise, we are given that the x-intercept is -10. This corresponds with the point (-10,0). We can substitute this information into the given equation to solve for our coefficient of y. 22​ x+22​ y=30y=0, x=-1022​ (-10)+22​⋅0=30Zero Property of Multiplication22​ (-10)=30LHS/(-10)=RHS/(-10)22​=-3 This means that the coefficient for x is -3. Our equation is now: -3x+22​ y=30.y-intercept To find the coefficient for y, we will go through a similar process. We are given that the y-intercept is the point (0,5). We can substitute this point into the equation and solve. -3x+22​ y=30x=0, y=5-3⋅0+22​ 5=30Zero Property of Multiplication22​ 5=30LHS/5=RHS/522​=6 Hence, the full equation is: -3x+6y=30.
Exercises 36 The standard form of a linear equation is written as: Ax+By=C. When we want to find the x-intercept, we substitute y with 0 and solve for x. Ax+By=Cy=0Ax+B⋅0=C Solve for x MultiplyAx=CLHS/A=RHS/A x=AC​ Similarly, finding the y-intercept means we have to substitute x with 0 and solve for y. Ax+By=Cx=0A⋅0+By=C Solve for y MultiplyBy=CLHS/B=RHS/B y=BC​ Therefore, the x- and y-intercepts of any line written in standard form are: x=AC​andy=BC​. Here we notice that in order to have integer values for intercepts we need to choose a value for C such that it is divisible by both A and B. The simplest case will be: C=A⋅B. This number is a multiple of A and B and will therefore be divisible by both of those coefficients. Now we have the equation in the form of Ax+By=AB. Let's choose A=2 and B=3 and substitute this into the formula above. Ax+By=ABA=2, B=32x+3y=2⋅3Multiply2x+3y=6
Exercises 37 The standard form for a linear equation in two variables, x and y, is written as: Ax+By=C, where A, B, and C are constants. Let's look at how we can relate this to both horizontal and vertical lines.Horizontal lines The standard form for a horizontal line is y=b, where b is any real number. How can we make y=b look like Ax+By=C? What coefficient for x would cause x to be eliminated from the final equation? Let's recall the Zero Property of Multiplication: x⋅0=0. Let's see what happens if we substitute A=0, B=1, and C=b into the standard form for a linear equation. Ax+By=CSubstitute values0⋅x+1⋅y=bZero Property of Multiplication1y=bIdentity Property of Multiplicationy=b Therefore, we can see that the equation of a horizontal line is simply the standard form for a linear equation with A=0 and B=1.Vertical lines We can go through a similar process for a vertical line. The standard form for a vertical line is x=a, where a is any real number. How can we make x=a look like Ax+By=C? Let's see what happens if we substitute A=1, B=0, and C=a into the standard form for a linear equation. Ax+By=CSubstitute values1⋅x+0⋅y=aZero Property of Multiplication1x=aIdentity Property of Multiplicationx=a Very similar to the equation for a horizontal line, the equation for a vertical line is just the standard form for a linear equation with A=1 and B=0.
Exercises 38 The x- and y-intercepts are the points where a function crosses the x- and y-axes. To find the x-intercept, you need to find the value of x when y=0, and vice versa for the y-intercept. We can solve for the intercepts in terms of k first. 3x+5y=k Solve for x-intercept y=03x+5⋅0=kZero Property of Multiplication3x=kLHS/3=RHS/3 x=3k​ The x-intercept of the equation is x=3k​. Since we know that the intercepts are integers, k must be divisible by 3. 3x+5y=k Solve for y-intercept x=03⋅0+5y=kZero Property of Multiplication5y=kLHS/35=RHS/35 y=5k​ The y-intercept of the equation is y=5k​. Because this intercept must also be an integer, k must be divisible by 5 as well. Knowing that k must be divisible by both 3 and 5, we can say that it must be divisible by their least common multiple, 15. Therefore, k is any number that divides evenly by 15.
Exercises 39 To simplify this expression, we need to consider the order of operations, We should fully simplify the numerator and denominator before trying to calculate the quotient. 4−(-4)2−(-2)​a−(-b)=a+b4+42+2​Add terms84​ba​=b/4a/4​21​ ⧼ebox-type-a-different-point-of-view...⧽ info How to understand subtracting a negative number You might get confused about what happens when you subtract a negative number. a−(-b)=a+b Try thinking about it in terms of English instead. When you use a double negative in conversation, it changes the meaning to be positive. Your friend: Why are you going to the school play this Saturday? You: I would never NOT go to a school play! I get extra credit in my English class if I go! Saying "never not" means you always go. Math works the same way!
Exercises 40 To simplify this expression, we need to consider the order of operations. We should fully simplify the numerator and denominator before trying to calculate the quotient. 0−214−18​Subtract terms-2-4​-b-a​=ba​2 ⧼ebox-type-a-different-point-of-view...⧽ info How to understand dividing two negative numbers You might get confused about changing the sign when dividing two negative numbers. -b-a​=ba​ Try thinking about it in terms of English instead. When you use a double negative in conversation, it changes the meaning to be positive. Your friend: Why are you going to the school play this Saturday? You: I would never NOT go to a school play! I get extra credit in my English class if I go! Saying "never not" means you always go. Math works the same way!
Exercises 41 To simplify this expression, we need to consider the order of operations. We should fully simplify the numerator and denominator before trying to calculate the quotient. 8−(-7)-3−9​a−(-b)=a+b8+7-3−9​Add and subtract terms15-12​ba​=b/3a/3​5-4​Put minus sign in front of fraction-54​ ⧼ebox-type-a-different-point-of-view...⧽ info How to understand subtracting a negative number You might get confused about what happens when you subtract a negative number. a−(-b)=a+b Try thinking about it in terms of English instead. When you use a double negative in conversation, it changes the meaning to be positive. Your friend: Why are you going to the school play this Saturday? You: I would never NOT go to a school play! I get extra credit in my English class if I go! Saying "never not" means you always go. Math works the same way!
Exercises 42 To simplify this expression, we need to consider the order of operations. We should fully simplify the numerator and denominator before trying to calculate the quotient. -5−(-2)12−17​a−(-b)=a+b-5+212−17​Add and subtract terms-3-5​-b-a​=ba​35​ ⧼ebox-type-a-different-point-of-view...⧽ info How to understand subtracting a negative number You might get confused about what happens when you subtract a negative number. a−(-b)=a+b Try thinking about it in terms of English instead. When you use a double negative in conversation, it changes the meaning to be positive. Your friend: Why are you going to the school play this Saturday? You: I would never NOT go to a school play! I get extra credit in my English class if I go! Saying "never not" means you always go. Math works the same way!
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