Math Core

Some linear relationships come out naturally as a total: "adult tickets cost $8, child tickets cost $5, and we sold $200 worth." Neither quantity is the "input." Standard form writes a line with $x$ and $y$ side by side, and it makes the intercepts especially easy to find.

The form

Definition

Standard form

The standard form of a linear equation is

Ax + By = C,

where $A$ , $B$ and $C$ are integers, and $A$ and $B$ are not both zero. By convention, $A$ is not negative, and $A$ , $B$ and $C$ have no common factor other than $1$ .

So $2x + 3y = 12$ and $4x - y = 7$ are in standard form. The equation $y = 2x + 5$ is not (the $x$ -term is on the wrong side), and neither is $\dfrac{1}{2}x + y = 3$ (the coefficient is a fraction).

The ticket example becomes $8x + 5y = 200$ , where $x$ is the number of adult tickets and $y$ the number of child tickets. Each term is money from one kind of ticket, and they add up to the total. Many "combination" problems look like this.

Standard form can also describe lines that slope-intercept form can't. The vertical line $x = 4$ is $1x + 0y = 4$ , which fits $Ax + By = C$ with $B = 0$ .

Intercepts: the fast way to graph

The $x$ -intercept is where a line crosses the $x$ -axis. Every point there has $y = 0$ . Likewise, every point on the $y$ -axis has $x = 0$ .

Finding intercepts

To find the $x$ -intercept, set $y = 0$ and solve for $x$ .
To find the $y$ -intercept, set $x = 0$ and solve for $y$ .

Plot both intercepts and draw the line through them.

In standard form, setting one variable to $0$ makes its whole term vanish, so each intercept takes one step.

Worked example: Graphing with intercepts

Graph $2x + 3y = 12$ .

$x$ -intercept: $2x + 3(0) = 12$ , so $2x = 12$ and $x = 6$ . The point is $(6, 0)$ .
$y$ -intercept: $2(0) + 3y = 12$ , so $3y = 12$ and $y = 4$ . The point is $(0, 4)$ .

The graph of 2x + 3y = 12 crosses the axes at (6, 0) and (0, 4).Open in grapher →

Common mistake

Write intercepts as points with the $0$ in the right place. The $x$ -intercept $6$ is the point $(6, 0)$ , not $(0, 6)$ . The zero goes in the coordinate for the other axis.

Converting between forms

From standard form to slope-intercept form, solve for $y$ . For $2x + 3y = 12$ :

3y = -2x + 12 \quad\Longrightarrow\quad y = -\frac{2}{3}x + 4.

The slope is $-\dfrac{2}{3}$ , which matches the graph: from $(0, 4)$ down $4$ and right $6$ to $(6, 0)$ , and $\dfrac{-4}{6} = -\dfrac{2}{3}$ .

Doing the same thing to $Ax + By = C$ in general gives $y = -\dfrac{A}{B}x + \dfrac{C}{B}$ , so

\text{slope} = -\frac{A}{B} \qquad (B \ne 0).

From slope-intercept form to standard form, move the $x$ -term to the left side, clear any fractions, and make $A$ positive.

Worked example: Converting to standard form

Write each equation in standard form.

$y = 4x - 7$
$y = -\dfrac{2}{3}x + 5$

Solutions.

Subtract $4x$ : $-4x + y = -7$ . Multiply every term by $-1$ so that $A$ is positive: $4x - y = 7$ .
Multiply every term by $3$ to clear the fraction: $3y = -2x + 15$ . Add $2x$ : $2x + 3y = 15$ .

Tip

When you multiply to clear fractions or fix a sign, multiply every term on both sides. Then check a point. For item 2, $(0, 5)$ is on the original line, and $2(0) + 3(5) = 15$ , so the standard form is right.

Standard form in context

Worked example: Selling tickets

A school play earns $200 from ticket sales. Adult tickets cost $8 and child tickets cost $5. Write an equation, find the intercepts, and explain what they mean. If $15$ adult tickets were sold, how many child tickets were sold?

Let $x$ = adult tickets and $y$ = child tickets: $8x + 5y = 200$ .

$x$ -intercept: $8x = 200$ , so $x = 25$ . If only adults came, $25$ tickets were sold.
$y$ -intercept: $5y = 200$ , so $y = 40$ . If only children came, $40$ tickets were sold.

For $x = 15$ : $8(15) + 5y = 200$ , so $120 + 5y = 200$ , $5y = 80$ and $y = 16$ child tickets.

In a situation like this, only points with whole-number coordinates that are $0$ or more make sense, because you can't sell $2.5$ tickets. The line still organizes all the possibilities.

Practice

Practice 1

Find the $x$ -intercept of $3x - 4y = 24$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find the $y$ -intercept of $3x - 4y = 24$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which is $y = 3x + 5$ written in standard form?

Practice 4

Which is $y = \dfrac{1}{2}x - 3$ written in standard form?

Practice 5

What is the slope of the line $5x + 2y = 8$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Rewrite $4x + 2y = 10$ in slope-intercept form, $y = mx + b$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

Which equation matches the line shown?

y = 2x + 6(-3, 0)(0, 6)Open in grapher →

Practice 8

Almonds cost $6 per pound and raisins cost $4 per pound. Maya spends exactly $48 on $x$ pounds of almonds and $y$ pounds of raisins, so $6x + 4y = 48$ . If she buys $3$ pounds of raisins, how many pounds of almonds does she buy?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.