Linear equations in two variables
Budgets, intercepts, standard form and writing the equation of any line. You will move easily between an equation, its graph and the story behind it.
Every solution is a point
An equation like 2x + 3y = 12 has two unknowns, so it has many solutions, not just one. Each solution is a pair (x, y), and every one of them is a point on the same straight line. Drag the lens to hunt for pairs that make the equation true.
Constraints: the budget line
Two-variable equations often describe a constraint. T-shirts cost $8, caps cost $6, and you must spend exactly $48. If x is t-shirts and y is caps, the constraint is 8x + 6y = 48. Neither variable is “the input”: they share one budget.
Intercepts in seconds
The x-intercept is where the line meets the x-axis, so y = 0 there. The y-intercept is where it meets the y-axis, so x = 0. For Ax + By = C, cover one term and solve what is left.
- x-intercept: cover the 4y (y = 0). 3x = 24, so x = 8. Point (8, 0).
- y-intercept: cover the 3x (x = 0). 4y = 24, so y = 6. Point (0, 6).
- Two points are enough to draw the whole line.
Common trapMixing up which variable to set to 0. On the x-axis, the y-value is 0. On the y-axis, the x-value is 0.
Standard form: Ax + By = C
To read the slope from Ax + By = C, rearrange into y = mx + b. The shortcut: slope = −A ÷ B and y-intercept = C ÷ B. Type your own A, B and C below and watch the steps.
Write the equation of a line
Whatever you are given, the plan is the same: find the slope, then substitute one point into y = mx + b to find b. Given two points, get the slope from the formula. Given a point and a slope, you already have m.
Another routePoint-slope form, y − y₁ = m(x − x₁), gives the same line. y − 7 = 3(x − 2) expands to y = 3x + 1.
Parallel and perpendicular lines
Parallel lines never meet, so they have the same slope. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign. Their slopes multiply to −1.
How one variable affects the other
With a fixed total, the two variables trade off against each other. In 8x + 6y = 48, one extra t-shirt uses $8, which is 8 ÷ 6 = 4/3 of a cap. A neat way to see it: the slope of the line is −4/3, so every step of 3 in x is a drop of 4 in y.
Match a table to an equation
When an answer choice gives an equation and the question gives a table, the fastest check is to substitute rows. An option must work for every row, so test at least two.
| x | y |
|---|---|
| −1 | 7 |
| 0 | 5 |
| 2 | 1 |
Spot the trap
Each solution below has one faulty step, and each slip is a real SAT wrong answer. Find it.
Desmos moves
Desmos is built into the Digital SAT. For two-variable equations it can save you rearranging entirely. Press play on each move.