SAT Math — systems of two linear equations

A pair of linear equations can do only three things. The lines cross once, and there is a single solution; they never cross, and there is none; or they lie exactly on top of each other, and every point on one is a solution. Most of what the Algebra domain asks about systems is one of those three situations dressed in different clothes, and a learner who holds the three cases clearly in mind can often answer without doing much arithmetic at all.

Three questions here are ordinary solving, chosen so that each one rewards a different method: a pair whose coefficients already cancel when the equations are added, a pair where one equation must be multiplied through first, and a pair where one variable is already isolated so that substitution is quicker than anything else. The wrong options are the values a learner actually reaches — the other variable, the line before the final division, the constants combined in the wrong direction.

Two questions ask for something other than a variable: the value of an expression like one combination of the two unknowns. These look harder and are usually easier, because the expression asked for is frequently one equation subtracted from the other, term by term. Noticing that saves the whole solve, and the explanations show the comparison that reveals it.

Four questions are about the structural cases. Two ask for the coefficient that makes a system have no solution, which needs proportional left-hand sides together with constants that refuse to follow — and the explanations insist on that second half, because matching the coefficients and stopping is precisely how a learner produces identical lines while intending parallel ones. Two ask for the constant that makes a system have infinitely many solutions, with the untouched constant sitting among the options as the trap. One of them asks not for the constant itself but for the sum of two constants, so that finding half the answer is not enough.

The remaining three are a system built from a situation in words, a sum-and-difference pair, and a question in which four candidate pairs are offered and only one satisfies both equations. That last one is built so that all four pairs satisfy the first equation, which makes the point that a system is a conjunction: a pair that passes one test and fails the other has not solved anything.

Every sum of money is written in words, every store and student is invented, and nothing is drawn from any official publication.

• Choose between elimination and substitution by looking at the coefficients before starting
• Answer a question that asks for an expression rather than a variable by combining the equations directly
• State the condition for a system to have no solution, including the part about the constants that is usually omitted
• State the condition for infinitely many solutions, and recognise the untouched constant as the classic wrong answer
• Build a system from a situation in words and then report the quantity actually asked for
• Test a candidate pair against both equations rather than against whichever one is checked first

Written for this catalogue, with no source document. The content is the systems material named in the College Board's own public description of the Algebra domain — systems of two linear equations in two variables, including the conditions for no solution and for infinitely many; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test also asks some questions in a student-produced response format, which a multiple-choice bank cannot reproduce. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.