United States of AmericaPA
High School Core Standards - Algebra I Assessment Anchors

# 4.08 Standard form

Lesson

There are a number of ways of stating an equation for a straight line. Previously, we saw slope-intercept form and we will see others in future lessons.  Now, we'll discover the value of writing equations in standard form.

The standard form

The standard form of a linear equation is

$Ax+By=C$Ax+By=C

where $A$A, $B$B, and $C$C are all integers and the value of $A$A is positive, that is, $A>0$A>0.

When we are given an equation in standard form, we can either graph using intercepts or rearrange it to one of our other forms.

### Graphing using intercepts

The standard form of a line is great for identifying both the $x$x and $y$y intercepts.

For example, the line $2x+3y=6$2x+3y=6

The $x$x intercept happens when the $y$y value is $0$0

 $2x+3y$2x+3y $=$= $6$6 $2x+3\left(0\right)$2x+3(0) $=$= $6$6 $2x$2x $=$= $6$6 $x$x $=$= $3$3

The $y$y intercept happens when the $x$x value is $0$0.

 $2x+3y$2x+3y $=$= $6$6 $2\left(0\right)+3y$2(0)+3y $=$= $6$6 $3y$3y $=$= $6$6 $y$y $=$= $2$2
From here, we just need to plot the $x$x intercept $3$3, and the $y$y intercept $2$2, and draw the line through both.

### Rearranging to slope-intercept form

If we don't want to graph using intercepts, we can also rearrange the equation to slope-intercept form. Let's look at rearranging between the two forms.

#### Worked example

Express the equation $4x+6y=12$4x+6y=12 in slope-intercept form.

Think: We need to solve for $y$y to get the equation to the form $y=mx+b$y=mx+b.

Do: To solve for $y$y, we need to reverse the operations to get $y$y by itself.

 $4x+6y$4x+6y $=$= $12$12 Start with the given equation $6y$6y $=$= $-4x+12$−4x+12 Undo addition by subtracting $4x$4x from both sides $y$y $=$= $\frac{-4x+12}{6}$−4x+126​ Undo multiplication by dividing by $6$6 on both sides $y$y $=$= $\frac{-2}{3}x+2$−23​x+2 Simplify

Reflect: $y$y is isolated, so the linear equation $4x+6y=12$4x+6y=12 is $y=\frac{-2}{3}x+2$y=23x+2 in slope-intercept form.

### Rearranging to standard form

As a mathematical convention, we are often asked to give our answer in standard form. Let's look at rearranging to standard form.

#### Worked example

Express the equation $y=\frac{-3}{4}x-5$y=34x5 in standard form.

Think: We need get to the form $Ax+By=C$Ax+By=C, where $A$A, $B$B and $C$C are integers and $A>0$A>0.

Do: We need to ensure there are no non-integer coefficients and that $A$A is positive.

 $y$y $=$= $\frac{-3}{4}x-5$−34​x−5 Start with the given equation $4y$4y $=$= $-3x-20$−3x−20 Clear the fraction by multiplying by $4$4 $3x+4y$3x+4y $=$= $-20$−20 Add $3x$3x to both sides to get the x and y terms to the same side

Reflect: We now have something of the form $Ax+By=C$Ax+By=C, where $A$A$B$B and $C$C are integers and$A>0$A>0, so the linear equation $y=\frac{-3}{4}x-5$y=34x5 is $3x+4y=-20$3x+4y=20 in standard form.

#### Practice questions

##### question 1

Express the following equations in standard form.

1. $y=6x-5$y=6x5

2. $y=\frac{6x}{5}-6$y=6x56

##### question 2

Consider the line given by the equation: $5x-3y=-15$5x3y=15

1. Solve for $x$x-value of the $x$x-intercept of the line.

2. Solve for $y$y-value of the $y$y-intercept of the line.

3. Hence, graph the equation of the line.

##### question 3

A line has slope $\frac{7}{3}$73 and passes through the point $\left(7,-9\right)$(7,9).

1. By substituting into the equation $y=mx+b$y=mx+b, find the value of $b$b for this line.

2. Write the equation of the line in standard form.

### Outcomes

#### A1.2.1.2.1

Create, interpret, and/or use the equation, graph, or table of a linear function.

#### A1.2.1.2.2

Translate from one representation of a linear function to another (i.e., graph, table, and equation).

#### A1.2.2.1.3a

Write or identify a linear equation when given • the graph of the line Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

#### A1.2.2.1.3b

Write or identify a linear equation when given • two points on the line, or Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.

#### A1.2.2.1.3c

Write or identify a linear equation when given • the slope and a point on the line. Note: Linear equation may be in point-slope, standard, and/or slope-intercept form.