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SAT Math Unit 2: Linear functions & systems

Linear functions model constant rates. On the SAT you'll read slope and intercept in context and solve systems, often as word problems.

Covers: Slope and intercepts, graphs of linear equations, parallel lines, systems of two linear equations, interpreting models

Study guide

A linear function has a constant rate of change, the slope m = (y₂ − y₁)/(x₂ − x₁). In slope-intercept form, y = mx + b, the slope tells how much y changes for each one-unit increase in x, and the y-intercept b is the value of y when x = 0. Standard form, Ax + By = C, is easy to graph using its intercepts: set x = 0 to find the y-intercept and y = 0 to find the x-intercept.

The SAT loves interpreting models. In a context like cost = 25x + 100, the slope (25) is the rate, such as dollars per month, and the intercept (100) is the starting amount, such as a one-time fee. Always attach units when you interpret.

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (their product is −1). A system of two linear equations can be solved by substitution (solve one equation for a variable and plug it in) or elimination (add or subtract the equations to cancel a variable). Graphically, the solution is the intersection point. Different slopes give exactly one solution; equal slopes with different intercepts give parallel lines and no solution; identical equations give infinitely many solutions.

A point is a solution to a system of linear inequalities only if it makes every inequality true, so substitute its coordinates into each one. Graphically, the solution set is the region where the shaded half-planes overlap; a solid boundary means the line is included (≤ or ≥) and a dashed one means it is not. Replacing f(x) with f(x) + k shifts the line up k units, and replacing it with f(x − h) shifts it right h units; both change the intercepts but not the slope. To find a new intercept, write the transformed equation and then set x = 0 or y = 0.

If a line has slope m and passes through (x₁, y₁), its equation is y − y₁ = m(x − x₁). You can also substitute the point into y = mx + b to solve for b.

Key ideas

• Slope = rise/run = change in y over change in x; in context, it's the rate.
• The y-intercept is the starting value (when x = 0).
• Parallel lines share a slope; perpendicular slopes are negative reciprocals.
• Solve systems by substitution or elimination. Desmos graphing works too.
• Systems ax + by = c and dx + ey = f have no solution when a/d = b/e ≠ c/f.

Formulas and rules

Common mistakes

• Swapping x and y differences in the slope formula.
• Interpreting the intercept as a rate.

Words to know

Slope formula

m = (y₂ − y₁)/(x₂ − x₁)

Y-intercept in context

The starting value: the output when x = 0 (e.g. a one-time fee).

Slopes of perpendicular lines

Negative reciprocals (product −1).

Inconsistent system (lines)

Same slope, different y-intercepts: parallel lines, no solution.

Elimination method

Add or subtract equations to cancel a variable.

Slope

Rate of change: (y₂ − y₁)/(x₂ − x₁).

Standard form (line)

Ax + By = C; find intercepts by setting x = 0 or y = 0.

Substitution method

Solve one equation for a variable and plug it into the other.

Parallel lines

Lines with equal slopes that never meet.

Interpreting slope in context

The change in output per one unit of input, with units (e.g. dollars per month).

Graphing a linear equation

Plot the y-intercept, then use the slope (rise over run) to find more points.

Sample practice questions

297 more questions like these, from level 1 to 5, are in the practice set.