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.
| Student Learning Objectives: |
|---|
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| | 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 of lines in standard form. Standard form can be used to find the slope and intercepts of a line, compare lines, and determine where lines intersect.
Catch-Up and Review
Here are some recommended readings before getting started with this lesson.
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Explore
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 standard form, A, B, and C are real numbers, but A and B cannot both be 0. Different values of A, B, and C can represent the same line. When possible, use the smallest whole-number values, with A being non-negative.
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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 Plot the intercepts in a coordinate plane.
3
Draw the Line Passing Through the Intercepts
expand_more Draw a line that passes 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 needs new school supplies.
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 purchased.
a
Find the intercepts of the line of the given equation.
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b
Graph the line.
c
Interpret the intercepts in the context.
Answer
a
x-intercept: 10
y-intercept: 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
To find the intercepts of the line, we set one variable equal to 0 and solve for the other. Let's start by finding the x-intercept by substituting 0 for y and solving 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
We found that when y=0, x=10, so the x-intercept is 10, or the point (10,0). Next, we find the y-intercept by substituting 0 for x and solving 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 4, or the point (0,4).
b
Now we want to graph the given line. In Part A, we found the intercepts, so we know two points on the line.
rc Equation: & 2x+5y = 20 [0.5em] x-intercept: & (10,0) [0.5em] y-intercept: & (0,4) We plot the intercepts (10,0) and (0,4) on a coordinate plane and connect them with a line. Recall that the number of items cannot be negative, so we only need to graph non-negative values of x and y.
c
Lastly, we interpret the intercepts of the line.
The x-intercept (10,0) means Tearrik can buy 10 pencils if he buys no notebooks. The y-intercept (0,4) means he can buy 4 notebooks if he buys no pencils. These intercepts show the two extreme ways he can spend all 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 in slope-intercept form. y = 4/5x + 2/3 The equation can be rewritten in standard form using the Properties of Equality. Ax+By=C Here, A, B, and C are real numbers and A and B cannot both be equal to 0. It is preferred that A, B, and C are the smallest possible integers and that A is non-negative.
1
Remove All Fractions
expand_more When a linear equation contains fractions, the first step is to remove them. Multiply the equation by the least common denominator of the fractions. In this equation, the least common denominator is the product of 5 and 3, or 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 Move the terms containing variables to the left-hand side of the equation and any constant terms to the right-hand side.
Since a positive coefficient for x is preferred, multiply the equation by - 1.
This equation is now in standard form. Notice that the values of A, B, and C are in their smallest possible integer forms and A is non-negative.
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Example
Moving Between Forms
Consider the equation written in slope-intercept form. y=4x-8
a
Write the equation in standard form.
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b 
Select the correct graph of the line.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
a
All x- and y-terms should be on one side of the linear equation and the constants should be on the other side in standard form.
b
Plot the intercepts and connect them with a line.
Solution
a
In standard form, all x- and y-terms are on one side of the linear equation, and the constant is on the other side. The coefficient of x should be positive.
y=4x-8 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. The coefficient of x should be positive when writing the equation in standard form.
We wrote the standard form of the equation. c|c Slope-Intercept Form of the Line & Standard From of the Line y=4x-8 & 4x-y=8
b
To graph the line, we can start by finding its intercepts. First, let's substitute 0 for y to find the x-intercept.
The x-intercept is (2,0). Next, we 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). Finally, we plot the intercepts on a coordinate plane and connect them to graph the line.
This matches graph B.
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Example
Comparing Prices
A science museum sells individual child tickets and family passes. The cost of 5 child tickets is $10 less than the cost of 2 family passes.
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
In slope-intercept form, the coefficient of x is the slope and the constant is the y-intercept.
b
Gather all variable terms on one side of the linear equation and the constant on the other side.
a
To find the equation relating the child ticket price to the family pass price, we will find the slope and the y-intercept of the line. We start by finding the y-intercept from the graph.
The y-intercept of the line is 5. Next, we find the ratio of the change in y-values to the change in x-values to find the slope.
As the x-values increase by 2, y-values increase by 5. Slope&=changes iny-value/changes inx-value [0.9em] &=5/2 The slope is the coefficient of x, and the y-intercept is the constant in the equation. Finally, we substitute these values into the equation to write the equation of the graph in slope-intercept form. y&= mx+ n [0.9em] &= 52x+ 5
b
Now we want to rewrite the equation of the given relationship in standard form. To do this, we rearrange the terms in the equation from Part A so that the variables are on one side and the constant is on the other. We use the smallest possible whole-number values for the coefficients, with the coefficient of x positive.
y=5/2x+5
SubEqn
LHS-y=RHS-y
0 = 5/2x + 5 - y
SubEqn
LHS-5=RHS-5
- 5 = 5/2 x - y
RearrangeEqn
Rearrange equation
5/2x - y = - 5
MultEqn
LHS * 2=RHS* 2
2(5/2x - y ) = 2(- 5)
Distr
Distribute 2
2 * 5/2x - 2y = 2(- 5)
DenomMultFracToNumber
2 * a/2= a
5x - 2y = 2(- 5)
Multiply
Multiply
5x - 2y = - 10
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Pop Quiz
Finding the Values of A, B, and C
The applet shows a line and its equation in slope-intercept form, along with its graph. Find the values of A, B, and C of the line in standard form. Use the smallest possible integers, with A greater than or equal to 0.
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Pop Quiz
Converting the Slope-Intercept Form of a Line to Standard Form
Rewrite the given equation in standard form.
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Closure
Exploring the Standard Form of a Line
This lesson explored the standard form of a line. Ax+By=C In this form, A, B, and C are real numbers. These properties for these numbers are preferred for equations written in standard form.
- A is non-negative.
- A and B cannot both equal 0.
- A, B, and C are the lowest possible integers.
Every line can be represented by exactly one equation in standard form when the equation meets these standards. However, we can write infinitely many equivalent equations that also represent that line.
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Standard Form of a Line
Exercises
Which of the following linear equations are in standard form? Select all that apply.
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We can recall the standard form of a linear equation. Ax+ By= C In this form, A, B, and C are real numbers and A and B cannot both equal 0. With this information, we can conclude that two of the given equations are written in standard form. 7x+ 5y&= 6 8x+ 9y&= 40
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Solution
The following equation is written in slope-intercept form. y=2/4x-5 Rewrite the equation in standard form.
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We want to rewrite the equation in standard form. Let's first recall the standard form of a line. Ax+ By= C In this form, A, B, and C are real numbers and A and B cannot both equal 0. Remember that the value of A should be positive. Let's rearrange the equation in standard form.
Standard form of the given equation is 2x-4y=20.
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Solution
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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":"y=","formTextAfter":null,"answer":{"text":["-3"]}}
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We substitute 2 for y. Then we can solve the equation for x by isolating it.
When y is 2, x is equal to 4.
We will substitute 4 for x and solve the equation for y.
The value of y is -3.
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Solution
Choose the correct option that shows the x- and y-intercepts of the given equation. 13x-2y=52
{"type":"choice","form":{"alts":["x=4, y=-26","x=13, y=-2","x=-2, y=13","x=-26, y=4"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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Remember that the x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The ordered pair of the point where the line crosses the x-axis is denoted as ( x, 0). This is why we substitute 0 for y and solve the equation for x to find the x-intercept.
The equation intercepts the x-axis at x=4. Similar to finding the x-intercept, the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. The ordered pair of the point where the line crosses the y-axis is denoted as ( 0, y). This is why we substitute 0 for x and solve the equation for y to find the y-intercept.
The equation intercepts the y-axis at y=-26.
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Solution
Take a look at the following equation that is written in standard form. 4x+5y=20 Which of the graphs correctly represents this equation?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's begin by finding the x- and the y-intercepts of the linear equation. 4x+5y=20 We will substitute 0 for y and solve the equation for x to find the x-intercept.
Finding the x-Intercept
The x-intercept of a line is the x-coordinate of the point where the line crosses the x-axis. The y-coordinate of the point is equal to 0. This means that we should substitute 0 for y to find the x-intercept of the given equation. Then, solve for x.
An x-intercept of 5 means that the graph passes through the x-axis at the point ( 5,0).
Finding the y-Intercept
Similarly, the y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis. The x-coordinate of the point at the y-intercept is 0. Therefore, we will substitute 0 for x to find the y-intercept.
A y-intercept of 4 means that the graph passes through the y-axis at the point (0, 4).
Graphing the Equation
We found the x- and y- intercepts of the equation.
| x-intercept | (5,0) |
|---|---|
| y-intercept | (0,4) |
We can now graph the equation by plotting the intercepts and connecting them with a line.
This graph corresponds to option A.
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Solution
Emily loves to stay active and challenge herself with physical activities. She decided to go on an adventure where she would both bike and run a total distance of 80 kilometers. Emily sets a goal for herself to bike for 5 hours and run for 3 hours. This real world situation is modeled by the following equation in standard form. 5x+3y=80 In this equation, Emily's speed while biking is represented by x and Emily's speed while running is represented by y. Find Emily's speed while biking if her speed while running is 5 kilometers per hour.
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We model the relationship between time and distance for Emily's adventure by using the standard form of a line. 5x+3y=80 We are given that Emily's speed while running is 5 kilometers per hour. We substitute 5 for y — the variable that represents Emily's speed while running.
Emily's speed while biking is 13 kilometers per hour.
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Solution
Standard Form of a Line
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