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.