SAT Algebra — systems of equations in real contexts

Word problems make up a large share of SAT Math, and in the Algebra domain many of them are systems of two linear equations in disguise. The difficulty is rarely the elimination itself; it is choosing the right two equations from the words and then saying what the solution means. This material trains both halves on situations adults actually meet.

You set up and solve a break-even problem in which revenue has to cover fixed costs plus a cost for each item, and compare two subscription plans, one with a sign-up fee and a lower monthly price and one without, to find when they cost the same. A graph of the two plans asks what their crossing point means, and a later month asks which plan is cheaper and by how much. You also meet two plans with the same hourly rate and different fees, whose costs never become equal: a system with no solution, seen in context.

Mixture and blend problems use the same two-equation structure: one equation for the total amount and one for the total salt or the total value. Further contexts are ticket sales at two prices, the hours worked at two part-time jobs, a boat traveling with and against a current, and a "twice as many" relation that is easy to write backward. Two questions ask for interpretation only: what the solution of a given system represents, and what a coefficient in a car-rental cost model means.

The quiz has twelve multiple-choice questions with worked explanations and a check of each answer against the original conditions. Distractors come from the usual traps: one rate used instead of the difference of rates, the other variable reported, a price per pound written where a total value belongs, the ratio reversed. The flashcards summarize the standard setups and the meaning of intersections, slopes and constants. A calculator is allowed throughout the SAT Math section.

The material is based on the College Board's published description of the SAT Math section (Algebra domain: systems of two linear equations in two variables, including creating and interpreting them in context). It is independent practice and is not produced or endorsed by the College Board.

• Translate a real situation into a system of two linear equations, choosing one equation for amounts and one for totals.
• Solve break-even and price-plan problems and decide which option is cheaper for a given input.
• Interpret the intersection of two linear models, and the solution of a system, in context.
• Recognize from equal rates and different fixed amounts that two models never agree.
• Set up mixture, blend, ticket, work-hour and current problems and interpret a coefficient in a cost model.

Practice material written by Zestly, based on the College Board's description of the digital SAT Math section, Algebra domain (skill: systems of two linear equations in two variables — create, solve and interpret in context).