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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.**

Explore

The applet shows two different equations for the same line.

Explore this applet by considering the following questions.

- What are the similarities and differences between the equations?
- How do the coefficients of the equations change as the line changes?

Discussion

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.

Pop Quiz

The given linear equation shows the relationship between the variables $x$ and $y.$ Determine if the equation is written in standard form.

Discussion

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.
*expand_more*
*expand_more*
*expand_more*
### Extra

General Formulas for the Intercepts of an Equation in Standard Form

$3x+5y=30 $

The graph of this function can be drawn in two steps.
1

Find the Intercepts

Begin by substituting $y=0$ to find the $x-$intercept of the equation.
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 $y.$
The $y-$intercept is $(0,6).$

$3x+5y=30$

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

$LHS/3=RHS/3$

$x=10$

$3x+5y=30$

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

$LHS/5=RHS/5$

$y=6$

$Equation:x-intercept:y-intercept: 3x+5y=30(10,0)(0,6) $

2

Plot the Intercepts

Now it is time to plot the intercepts in a coordinate plane.

3

Draw the Line Passing Through the Intercepts

Lastly, draw a line passing through these points.

Note 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 |
---|---|---|

$A =0,$ $B =0$ | $(AC ,0)$ | $(0,BC )$ |

$A=0,$ $B =0$ | The line is horizontal, $y=BC ,$ so it does not cross the $x-$axis. | $(0,BC )$ |

$A =0,$ $B=0$ | $(AC ,0)$ | The line is vertical, $x=AC ,$ so it does not cross the $y-$axis. |

Example

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.

a $x-intercept:$ $(10,0)$

$y-intercept:$ $(0,4)$

$y-intercept:$ $(0,4)$

b

c See solution.

b Plot the intercepts and connect them with a line segment.

c Remember that the number of pencils and notebooks cannot be negative.

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

$LHS/2=RHS/2$

$x=10$

$2x+5y=20$

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

$LHS/5=RHS/5$

$y=4$

b When graphing the line, the intercepts found in Part A can be used.

$Equation:x-intercept:y-intercept: 2x+5y=20(10,0)(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.

Discussion

Any linear equation can be rewritten in standard form. Consider the following linear equation that is written in slope-intercept form.
*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.
*expand_more*
*expand_more*

$y=54 x+32 $

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 $B$ 1

Remove All Fractions

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=54 x+32 $

MultEqn

$LHS⋅15=RHS⋅15$

$15y=(54 x+32 )15$

▼

Simplify right-hand side

Distr

Distribute $15$

$15y=54 x⋅15+32 ⋅15$

CommutativePropMult

Commutative Property of Multiplication

$15y=54 ⋅15⋅x+32 ⋅15$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$15y=54⋅15 x+32⋅15 $

ReduceFrac

$ba =b/5a/5 $

$15y=14⋅3 x+32⋅15 $

ReduceFrac

$ba =b/3a/3 $

$15y=14⋅3 x+12⋅5 $

DivByOne

$1a =a$

$15y=4⋅3x+2⋅5$

Multiply

Multiply

$15y=12x+10$

2

Rearrange the Terms of the Equation

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.

Example

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.
### Hint

### Solution

The standard form of the equation is written.
The $x-$intercept is $(2,0).$ Next, find the $y-$intercept of the line by substituting $0$ for $x$ in the equation.
The $y-$intercept is $(0,-8).$ Plot the intercepts on a coordinate plane. Then connect them with a line to graph the line.

$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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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.

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

$4x−y=8$

$Slope-InterceptForm of a Line y=4x−8 Standard Fromof a Line 4x−y=8 $

b The intercepts of a line are used to graph the line. First, find the $x-$intercept by substituting $0$ for $y.$

$4x−y=8$

Substitute

$x=0$

$4(0)−y=8$

▼

Solve for $y$

ZeroPropMult

Zero Property of Multiplication

$0−y=8$

SubTerms

Subtract terms

$-y=8$

MultEqn

$LHS⋅(-1)=RHS⋅(-1)$

$y=-8$

Example

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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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.$$SlopeSlope =changes inx-valuechanges iny-value ⇓=25 $

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.
$yy =mx+n⇓=25 x+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.
$y=25 x+5$

MultEqn

$LHS⋅2=RHS⋅2$

$2y=2⋅25 x+10$

▼

Multiply

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$2y=22⋅5 x+10$

CancelCommonFac

Cancel out common factors

$2y=2 2 ⋅5 x+10$

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

$5x−2y=-10$

$5x−2y=-10 $

Pop Quiz

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. Pop Quiz

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.