How to solve SAT linear equations in two variables

A linear equation in two variables, such as $14a + 9c = 1260$ or $y = -\frac{3}{2}x + 8$, describes every pair of values that satisfies a condition. Its graph is a line. On the Digital SAT this skill covers writing these equations from word problems, reading slope and intercepts from different forms, and working with parallel and perpendicular lines. Algebra is about 35% of the Math section, or 13 to 15 of the 44 questions, and roughly 2 to 4 of those usually test this skill.

What the questions look like

Model a situation. A store sells two products at different prices and takes in a fixed total, or a person splits time between two activities. You choose or write the equation, find one variable when the other is given, or explain what a coefficient or intercept means.

Find the line. You are given a graph, two points, or a slope and a point, and you choose the matching equation. The choices often mix slope-intercept form $y = mx + b$, standard form $Ax + By = C$, and point-slope form $y - y_1 = m(x - x_1)$.

Parallel and perpendicular lines. You are told one line's equation and asked for a line that is parallel or perpendicular to it through a given point. Some questions ask which value of a constant makes two lines parallel.

Step-by-step method

1. In a word problem, pair each rate with its variable. If adult tickets cost $14 and $a$ is the number of adult tickets, the adult revenue is $14a$. Add the pieces and set them equal to the total.
2. Get the slope from any form. In $y = mx + b$ the slope is $m$. In $Ax + By = C$ the slope is $-\dfrac{A}{B}$. With two points, use $\dfrac{y_2 - y_1}{x_2 - x_1}$.
3. Find intercepts by setting a variable to 0. Set $x = 0$ to get the $y$-intercept. Set $y = 0$ to get the $x$-intercept. In context, an intercept is the value of one quantity when the other is zero.
4. Use slope rules for related lines. Parallel lines have equal slopes and different intercepts. Perpendicular lines have slopes that multiply to $-1$, so the new slope is the negative reciprocal.
5. Check with a point. Substitute a known point into the equation you chose. If both sides match, the point is on the line.

Worked example

Example 1. A theater sells adult tickets for $14 each and child tickets for $9 each. On one night it collected $1,260 from $a$ adult tickets and $c$ child tickets.

(a) Write an equation relating $a$ and $c$. (b) If 70 child tickets were sold, how many adult tickets were sold? (c) What does the $a$-intercept of the graph represent?

(a) Revenue from adults plus revenue from children equals the total:

$$14a + 9c = 1260$$

(b) Substitute $c = 70$: $14a + 630 = 1260$, so $14a = 630$ and $a = 45$.

(c) The $a$-intercept is where $c = 0$: $14a = 1260$, so $a = 90$. It means the theater would have needed to sell 90 adult tickets to collect $1,260 if it sold no child tickets.

Example 2. Line $\ell$ has equation $4x - 6y = 15$. Line $k$ is perpendicular to $\ell$ and passes through $(2, 5)$. What is an equation of line $k$?

The slope of $\ell$ is $-\dfrac{A}{B} = -\dfrac{4}{-6} = \dfrac{2}{3}$.

The perpendicular slope is the negative reciprocal, $-\dfrac{3}{2}$. Use point-slope form with $(2, 5)$:

$$y - 5 = -\frac{3}{2}(x - 2)$$

Distribute: $y - 5 = -\frac{3}{2}x + 3$, so $y = -\frac{3}{2}x + 8$.

If the choices are in standard form, multiply by 2 and rearrange: $3x + 2y = 16$. Check with the point: $3(2) + 2(5) = 16$. It works.

Common traps

• Reading the slope straight off standard form. In $4x - 6y = 15$, the slope is not $4$, and it is not $-\frac{2}{3}$. Use $-\frac{A}{B}$ to get $\frac{2}{3}$, or solve for $y$ if you are unsure.
• Taking only half of "negative reciprocal." The perpendicular slope to $\frac{2}{3}$ is $-\frac{3}{2}$. Both $\frac{3}{2}$ and $-\frac{2}{3}$ appear as wrong choices.
• Attaching a price to the wrong variable. If the question defines $a$ as adult tickets, the $14 goes with $a$. Choices that swap the coefficients look right at a glance.
• Misreading an intercept in context. The $a$-intercept in Example 1 is not "the number of adult tickets sold." It is the number of adult tickets that would produce the full total with zero child tickets.
• Swapping coordinates. A point is written $(x, y)$. When you check $(2, 5)$, $x$ is 2 and $y$ is 5. Substituting them backward is a fast way to reject the right answer.

Desmos tip

Desmos graphs equations in any form, so you can type $4x - 6y = 15$ without rearranging it. Click the line to see its intercepts. When the answer choices are four equations, graph the ones that look plausible and see which passes through the given points. If a question has an unknown constant, type the equation with the constant and use the slider to see how the line moves.