3. Standard Form of a Line
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Chapter 7
3.
Standard Form of a Line
Understanding the standard form of a line is crucial for interpreting and graphing linear equations effectively. This approach offers a structured way to represent relationships between variables, which is particularly useful in solving problems that involve multiple constraints. Rewriting equations into this format allows to simplify graphing and analysis tasks, making it easier to work with linear models in real-world scenarios such as economics, engineering, and data representation.
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Lesson Settings & Tools
| | 12 Theory slides |
| | 12 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Standard Form of a Line
Slide of 12
This lesson focuses on writing equations in standard form. Standard form is a special way to write equations for lines. It gives us important information about the slopes of the lines, intercepts, and how it relates to other lines. Understanding the standard form helps to solve equations, analyze systems of lines, and even predict where lines will meet or run parallel.
Catch-Up and Review
Here are some recommended readings before getting started with this lesson.
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Comparing the Slope-Intercept Form and Standard Form
The applet shows two different equations for the same line.
- What are the similarities and differences between the equations?
- How do the coefficients of the equations change as the line changes?
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Discussion
Standard Form of a Line
In the standard form of a line all x- and y-terms are on one side of the linear equation or function and the constant is on the other side.
Ax+By=C
In this form, A, B, and C are real numbers. It is important to know that A and B cannot both be 0. Different combinations of A, B, and C can represent the same line on a graph. It is preferred to use the smallest possible whole numbers for A, B, and C and it is also better if A is a positive number.
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Pop Quiz
Recognizing the Standard Form of a Line
The given linear equation shows the relationship between the variables x and y. Determine if the equation is written in standard form.
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Discussion
Graphing a Linear Equation in Standard Form
A linear function written in standard form has quickly identifiable x- and y-intercepts. Since two points determine a line, this provides enough information to graph the function. Consider the following linear equation written in standard form. 3x+5y=30 The graph of this function can be drawn in two steps.
1
Find the Intercepts
expand_more Begin by substituting y= 0 to find the x-intercept of the equation.
3x+5y=30
Substitute
y= 0
3x+5( 0)=30
▼
Solve for x
ZeroPropMult
Zero Property of Multiplication
3x+0=30
IdPropAdd
Identity Property of Addition
3x = 30
DivEqn
.LHS /3.=.RHS /3.
x = 10
The x-intercept is (10,0). The y-intercept can be found in a similar way. Substitute x= 0 into the equation and solve for y.
3x+5y=30
Substitute
x= 0
3( 0)+5y=30
▼
Solve for y
ZeroPropMult
Zero Property of Multiplication
0+5y=30
IdPropAdd
Identity Property of Addition
5y = 30
DivEqn
.LHS /5.=.RHS /5.
y = 6
The y-intercept is (0,6). rc Equation: & 3x+5y = 30 x-intercept: & (10,0) y-intercept: & (0,6)
2
Plot the Intercepts
expand_more Now it is time to plot the intercepts in a coordinate plane.
3
Draw the Line Passing Through the Intercepts
expand_more Lastly, draw a line passing through these points.
Extra
General Formulas for the Intercepts of an Equation in Standard FormNote that general formulas for the intercepts can be derived for any linear function written in standard form Ax+ By= C.
| Assumption | x-intercept | y-intercept |
|---|---|---|
| A≠ 0, B≠ 0 | (C/A,0) | (0,C/B) |
| A= 0, B≠ 0 | The line is horizontal, y= C B, so it does not cross the x-axis. | (0,C/B) |
| A≠ 0, B= 0 | (C/A,0) | The line is vertical, x= C A, so it does not cross the y-axis. |
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Example
Stationery Shopping
Tearrik is excited to buy school supplies. He plans to buy some cool pencils and notebooks.
The pencils he wants cost $2 each, and the notebooks he likes cost $5 each. He has $ 20 to spend. The following linear equation models this situation. 2x+5y=20 In this equation, x represents the number of pencils and y represents the number of notebooks.
a Find the intercepts of the line whose equation is written in standard form.
b Graph the line.
c Interpret the intercepts in the context.
Answer
a x-intercept: (10,0)
y-intercept: (0,4)
y-intercept: (0,4)
b 
c See solution.
Hint
b Plot the intercepts and connect them with a line segment.
c Remember that the number of pencils and notebooks cannot be negative.
Solution
a The equation models the total cost of stationery that Tearrik buys. The number of pencils purchased is represented by x and the number of notebooks purchased is represented by x.
2x+5y=20 Begin by finding the x-intercept. Substitute 0 for y and solve the equation for x.
2x+5y=20
Substitute
y= 0
2x+5( 0)=20
▼
Solve for x
ZeroPropMult
Zero Property of Multiplication
2x+0=20
IdPropAdd
Identity Property of Addition
2x=20
DivEqn
.LHS /2.=.RHS /2.
x=10
The point (10,0) is the x-intercept. Next, the y-intercept will be found. Substitute 0 for x and solve for y.
2x+5y=20
Substitute
x= 0
2( 0)+5y=20
▼
Solve for y
ZeroPropMult
Zero Property of Multiplication
0+5y=20
IdPropAdd
Identity Property of Addition
5y=20
DivEqn
.LHS /5.=.RHS /5.
y=4
The y-intercept is (0,4).
b When graphing the line, the intercepts found in Part A can be used.
rc Equation: & 2x+5y = 20 [0.5em] x-intercept: & (10,0) [0.5em] y-intercept: & (0,4) Plot the intercepts on a coordinate plane. Then, connect the intercepts with a line.
The number of stationery purchased cannot be negative. In other words, stores do not sale a negative amount of products. This means that only positive values of x and y is considered in this context.
c Look at the intercepts in the graph to interpret them in the context.
First, the x-intercept means that if Tearrik does not buy any notebooks, he can buy 10 pencils with all of his money. Whereas, the y-intercept means that if Tearrik does not buy pencils, he can buy 4 notebooks using all of his money.
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Discussion
Writing a Linear Equation in Standard Form
Any linear equation can be rewritten in standard form. Consider the following linear equation that is written in slope-intercept form. y = 4/5x + 2/3 Using the Properties of Equality, the equation can be rewritten in standard form. Ax+By=C Here, A, B, and C are real numbers and A and B cannot both be equal to 0. It can be noted that representing A, B, and C with the smallest possible integers is preferred, as well as A being positive.
1
Remove All Fractions
expand_more When a linear equation contains fractions, the first step is to remove all fractions. The equation is multiplied by the least common denominator of the fractions. In this case, it is the product of 5 and 3, which is 15. Using the Multiplication Property of Equality, the equation can be written as follows.
y= 4/5x + 2/3
MultEqn
LHS * 15=RHS* 15
15y=(4/5x+2/3)15
▼
Simplify right-hand side
Distr
Distribute 15
15y=4/5x* 15+2/3* 15
CommutativePropMult
Commutative Property of Multiplication
15y=4/5* 15* x+2/3* 15
MoveRightFacToNum
a/c* b = a* b/c
15y=4* 15/5x+2* 15/3
ReduceFrac
a/b=.a /5./.b /5.
15y=4* 3/1x+2* 15/3
ReduceFrac
a/b=.a /3./.b /3.
15y=4* 3/1x+2* 5/1
DivByOne
a/1=a
15y=4* 3x+2* 5
Multiply
Multiply
15 y = 12x + 10
2
Rearrange the Terms of the Equation
expand_more Now, move the terms containing variables to the left-hand side of the equation. In addition to this, move the constant to the right-hand side using the Subtraction Property of Equality.
Since a positive coefficient for x is preferable, the equation can be multiplied by - 1.
This equation is now in the standard form. Note that the values of A, B, and C are in their smallest possible integer forms.
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Example
Preparing for School
Tearrik wanted to remember some mathematical contexts before going back to school. He wants to remember slope-intercept form and standard form of a line. This is why he wrote an equation in slope-intercept form. y=4x-8
a Help Tearrik to rewrite the equation in standard form.
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b Graph the line of the equation.
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Hint
a All x- and y-terms should be on one side of the linear equation and the constant should be on the other side in standard form.
b Plot the intercepts and connect them with a line.
Solution
a The equation that Tearrik wrote in slope-intercept form is given as follows.
y=4x-8 Remember that all x- and y-terms should be on one side of the linear equation and the constant should be on the other side in standard form. In addition to this, the coefficient of x should be positive when writing the equation in standard form.
y=4x-8
SubEqn
LHS-y=RHS-y
y-y=4x-8-y
SubTerms
Subtract terms
0=4x-8-y
AddEqn
LHS+8=RHS+8
0+8=4x-8-y+8
CommutativePropAdd
Commutative Property of Addition
0+8=4x-8+8-y
AddTerms
Add terms
8=4x-y
RearrangeEqn
Rearrange equation
4x-y=8
The standard form of the equation is written. c|c Slope-Intercept Form of a Line & Standard From of a Line y=4x-8 & 4x-y=8
b The intercepts of a line are used to graph the line. First, find the x-intercept by substituting 0 for y.
The x-intercept is (2,0). Next, find the y-intercept of the line by substituting 0 for x in the equation.
4x-y=8
Substitute
x= 0
4( 0)-y=8
▼
Solve for y
ZeroPropMult
Zero Property of Multiplication
0-y=8
SubTerms
Subtract terms
- y=8
MultEqn
LHS * (-1)=RHS* (-1)
y=-8
The y-intercept is (0,-8). Plot the intercepts on a coordinate plane. Then connect them with a line to graph the line.
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Example
Comparing Prices
Tearrik realizes at the stationery shop that the price of five pencil cases equals $ 10 less than the price of two school bags. The graph below shows the relationship between the price of the school bag and the price of the pencil case.
a Write the equation in slope-intercept form.
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b Rewrite the equation in standard form.
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Hint
a Remember that the coefficient of x is the slope, and the constant is the y-intercept in the slope-intercept form.
Solution
a The graph of the relationship between the price of a school bag and the price of the pencil case is given.
The y-intercept and the slope of the line is used to write the equation of the line in slope-intercept form. Let's first look at the y-intercept. Remember that the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis.
The y-intercept of the line is 5. Now find the ratio of the change in y-values to the change in x-values to find the slope.
When the x-values increase by 2, y-values increase by 5. Slope&=changes iny-value/changes inx-value &⇓ Slope&=5/2 The value of the slope is the coefficient of variable x, and the y-intercept is the constant of the equation.
Substitute the values into the equation to write the equation of the graph in slope-intercept form. y&= mx+ n &⇓ y&= 52x+ 5
b All terms for the variables x and y are on one side of the linear equation and the constant is on the other side in standard form.
Ax+By=C Additionally, the coefficients of the variables do not equal 0 in the standard form of an equation and, the coefficient of x is greater than 0. Rearrange the terms in the equation written in Part A to write it in standard form.
y= 52x+5
MultEqn
LHS * 2=RHS* 2
2y=2* 52x+10
▼
Multiply
MoveLeftFacToNum
a*b/c= a* b/c
2y= 2*52x+10
CancelCommonFac
Cancel out common factors
2y= 2*52x+10
SimpQuot
Simplify quotient
2y=5x+10
SubEqn
LHS- 2y+10=RHS- 2y+10
2y-( 2y+10)=5x+10-( 2y+10)
▼
Simplify equation
Distr
Distribute -1
2y-2y-10=5x+10-2y-10
CommutativePropAdd
Commutative Property of Addition
2y-2y-10=5x+10-10-2y
SubTerms
Subtract terms
- 10=5x-2y
RearrangeEqn
Rearrange equation
5x-2y=- 10
Note that the coefficients of the variables are 5 and -2 which are not equal to 0, and the coefficient of x is greater than 0. Here is the equation in standard form. 5x-2y=- 10
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Pop Quiz
Finding the Values of A, B, and C
The standard form of a linear equation is written as follows. Ax+By= C In general, A, B, and C are real numbers. It is preferred for A, B, and C to be the smallest possible integers and for A to be positive. Find the values of A, B, and C by using the given line that has an equation in slope-intercept form. In the answers, make sure that A, B, and C are the smallest possible integers and that A is positive.
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Converting the Slope-Intercept Form of a Line to Standard Form
The following equation displays the relationship between the variables x and y in slope-intercept form. Rewrite the equation in slope-intercept form in standard form.
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Closure
Exploring the Standard Form of a Line
In this lesson, the standard form of a line has been discussed, recognized, graphed, and applied to real-world examples. Ax+By=C In this form, A, B, and C are real numbers. The preferred properties for these numbers are as follows.
The number of equations in standard form that meets these properties is exactly one for each line. However, there are infinitely many equivalent equations in standard form. In other words, linear equations that describe the same line are equivalent. These equivalent equations can be written by using the Multiplication Property of Equality.
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Standard Form of a Line
Exercise 2.1
Kriz says the equation of the line given below is 2x+3y=18. LaShay says that the equation of the line is y=- 23x+6.
Who is correct?
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We want to decide who is correct about the equation of the given line. Notice that Kriz's equation is in standard form and LaShay's equation is in slope-intercept form.
| Equation | Form of the Equation | |
|---|---|---|
| Kriz | 2x+3y=18 | Standard Form |
| LaShay | y=- 23x+6 | Slope-Intercept Form |
We will write the equation of the given line in slope-intercept form and compare it with LaShay's equation. Additionally, we will convert the equation of the line to standard form and compare it with Kriz's equation. To start, we can find the slope of the line by selecting two points on the line and calculating the rise and run.
The rise is - 2 and the run is 3. Remember, the slope is the ratio of the rise to the run.
The slope of the line is - 23. Look at the graph to see the y-intercept of the line. The line intercepts the y-axis at (0,6). This means that the y-intercept is 6. We can write the equation of the line by substituting these values in slope-intercept form. y= mx+ b ⇓ y= -2/3x+ 6 The equation of the line in slope intercept form is y=- 23x+6. This means that LaShay is correct! Now we will convert the equation in standard form to see if Kriz is also right. We will write terms with the variables on one side of the equation and the constant on the other side. We will prefer the smallest possible integers for the coefficients.
The equation of the line in standard form is 2x+3y=18. This means that Kriz is also correct! We can conclude that both Kriz and LaShay are correct!
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Solution
Heichi goes grocery shopping with a total of $12 in his pocket. He discovers that one kilogram of oranges costs $3, while one kilogram of apples costs $2. He decides to buy both oranges and apples using his available money. The following equation models this situation. 3x+2y=12 In this equation, x is the number of kilograms of oranges and y represents the number of kilograms of apples. Graph the equation. What do the x- and the y-intercept represent in this context?
Choose the option that best describes the situation.
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We want to graph and interpret the x- and the y-intercept of the following equation. 3x+2x=12 The equation is written in standard form. We substitute 0 for x to find the y-intercept.
The y-intercept is 6. This means that the line of this equation intercepts the y-axis at (0,6). Now we can find the x-intercept by substituting 0 for y in the equation.
The line of this equation intercepts the x-axis at (4,0). Let's plot these points on the coordinate plane and graph a line that passes through these two points. Both the number of kilograms of orange and apple cannot be negative. This means that the line will be drawn only in the first quadrant of the coordinate plane.
Now we can interpret the x- and the y-intercepts. Keep in mind that the y-coordinate of the x-intercept is always 0. In this context, it represents the number of kilograms of oranges that can be purchased with the total amount of money.
Similarly, x-coordinate of y-intercept is 0. This can be interpreted as the amount of apples we can buy with our total money.
The best option to describe this situation is C.
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Solution
The graph below shows the time Ignacio spends to complete two projects for a class.
Here are some statements about the time spent on finishing these projects. A.& Ignacio will spend 180 minutes to complete & Project1if he decides to do only Project 1. B.& Ignacio completes two of & the projects in90minutes. C.& Ignacio will spend 40 minutes to complete & Project2if he completes Project1 in 50minutes. D.& Ignacio found that Project 1 is harder & than Project2. Which of these statements could be true?
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We want to decide which of the given statements could be true for the given graph. Notice that the points on the line of the equation satisfies the equation. The x-coordinate of the points represents the time spent on Project 1, and the y-coordinate of the points represents the time spent on Project 2.
Note that the sum of the coordinates of the points is the same, 90. This indicates that the required time to complete the projects is 90 minutes. We can write an equation using this information. x+ y=90 Let's look at the given statements to find the correct one.
A. Ignacio will spend 180 minutes to complete Project 1 if he decides to do only Project 1.
This statement means that Ignacio allocates all of his time to Project 1. This results in no time available for Project 2. We can substitute 0 for y in the equation to reflect this.
We found that x equals 90 minutes. This indicates that the time that we have to complete Project 1 is 90 minutes, not 180. This means that the first statement is false. We can continue with the second statement.
B. Ignacio completes two of the projects in 90 minutes.
We wrote the equation of the given graph. x+y=90 This equation shows that the total time spends to complete two of the project is 90 minutes, so this statement is correct. Let's look at the third option.
C. Ignacio will spend 40 minutes to complete Project 2 if he completes Project 1 in 50 minutes.
We will substitute 50 for the time spends to complete Project 1 into the equation to determine the allotted time for Project 2.
We found that y equals 40. We can conclude that Ignacio will complete Project 2 in 40 minutes if he spends 50 minutes to complete Project 1. We found that option C is correct. Take a look at option D.
D. Ignacio found that Project 1 is harder than Project 2.
We do not have enough information to determine if this statement is true, so this option is not the one we are looking for.
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Solution
Jordan has the option of working as a server and cook for a restaurant. She is paid $8 per hour while she works as a server. She is paid $10 per hour as a cook. Jordan is keeping track of how she earned $300 over her first few weeks.
Write an equation in standard form to represent the relationship between Jordan's working hours as a server and as a cook. In this equation, assign x as the number of hours she works as a server and y as the number of hours she works as a cook.
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We want to write an equation in standard form to represent Jordan's working hours as a cook and as a server. We have information about the hourly pay rates for both positions.
When we multiply the hourly pay by the number of working hours, we can determine the total amount of money earned from that job.
| Server | Cook | |
|---|---|---|
| Payment ($/h) | $ 8 | $ 10 |
| Number of Hour She Worked | x | y |
| Money She Earns | 8* x | 10* y |
We found the money she earned by working as a server and as a cook. Also, we want the total money she earns from working at the restaurant be $300. This means that when we add up the money that she earned as a server and as a cook, we should get a total of $300. We can write the equation in standard form. 8x +10y=300
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Standard Form of a Line
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