Digital SAT Algebra: a complete guide to linear equations, functions, systems and inequalities

What the Algebra domain demands

Algebra is the largest Digital SAT Math domain, about 35% of the section, and it is the most reliable place to bank points because the questions are formulaic once the skills are automatic. This guide ties together the matching dot-point pages, each with its own practice: linear equations in one variable, linear functions, linear equations in two variables, systems of two linear equations, and linear inequalities.

Linear equations in one variable

Solve by simplifying each side, collecting variable terms on one side and constants on the other, and dividing. Watch the two special cases: if the variable cancels and leaves a true statement (like $0 = 0$) there are infinitely many solutions; if it leaves a false statement (like $0 = 5$) there is no solution. Parameter questions ask which value forces one of these cases, so match coefficients and constants on both sides.

Linear functions and slope as a rate

A linear function $f(x) = mx + b$ has slope $m$ (the rate of change) and $y$-intercept $b$ (the starting value). When a question asks what a coefficient "represents", name the rate with its units and direction (increasing or decreasing). Build a model by reading the constant rate and the initial amount from the description.

Lines: the three forms

A line in two variables can be written three ways.

• Slope-intercept: $y = mx + b$ (read slope and intercept directly).
• Standard: $Ax + By = C$ (intercepts fall out by setting $y = 0$, then $x = 0$).
• Point-slope: $y - y_1 = m(x - x_1)$ (build from a slope and a point).

Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes (product $-1$).

Systems of two linear equations

A system is solved where the lines intersect. Use substitution when an equation is solved for a variable, elimination when adding or subtracting cancels a variable, or graph in Desmos and click the intersection. Count solutions by slope and intercept: different slopes give one solution, equal slopes with different intercepts give none, equal slopes with equal intercepts give infinitely many.

Linear inequalities

Solve like an equation, but flip the inequality sign whenever you multiply or divide by a negative. One-variable inequalities give a range; two-variable inequalities give a half-plane (solid boundary for $\le$ or $\ge$, dashed for $<$ or $>$, shade the side a test point satisfies). Word-problem constraints turn "at most" into $\le$ and "at least" into $\ge$.

How Algebra is examined

• Linear equations. Solve in one variable; identify no-solution and infinite-solution cases.
• Functions. Read slope as a rate and the intercept as a starting value; evaluate function notation.
• Lines. Move among slope-intercept, standard and point-slope; use parallel and perpendicular slopes.
• Systems. Solve by substitution, elimination or graphing; count solutions by slope and intercept.
• Inequalities. Solve with the sign-flip rule; graph half-planes; model constraints.

Check your knowledge

Work these under timed conditions, then read the solutions.

1. Solve $4(x - 1) = 2x + 6$ for $x$. (2 marks)
2. The line $5x - 2y = 10$ has what slope? (2 marks)
3. A line is perpendicular to $y = \frac{1}{4}x + 2$. What is its slope? (1 mark)
4. Solve the system $\begin{cases} x + y = 10 \\ x - y = 4 \end{cases}$. (2 marks)
5. Solve $-2x + 3 > 11$. (2 marks)