Solving pairs of linear equations by substitution or elimination, and reasoning about when systems have one, none, or infinitely many solutions.
59
Total questions
22
Easy
16
Medium
21
Hard
Systems questions come in two flavors: actually solving (find the intersection point, or the value of x + y at the solution), and reasoning about solution counts — a system has no solution when the lines are parallel (equal slopes, different intercepts) and infinitely many when the equations describe the same line.
Choose your weapon per question: elimination is usually fastest when coefficients align, substitution when one equation already isolates a variable, and Desmos when the question only needs the intersection point — graph both lines and read it off. For "no solution" questions with an unknown constant, set the slope ratios equal and solve for the constant directly.
Straight from the Grind1600 question bank — try each one before revealing the answer.
Correct answer: A
Choice A is correct. The second equation in the given system is a = 12. Substituting 12 for a in the first equation in the given system yields 6(12) - 5b = 7, or 72 - 5b = 7. Subtracting 72 from both sides of this equation yields -5b = -65. Dividing both sides of this equation by -5 yields b = 13. Therefore, the solution (a, b) to the given system of equations is (12, 13).
Choice B is incorrect and may result from adding 72 to 7 instead of subtracting, then dividing by 5 incorrectly.
Choice C is incorrect and may result from a sign error when dividing -65 by -5.
Choice D is incorrect and may result from adding 72 and 7 to get 79, then making additional calculation errors.
Correct answer: B
Choice B is correct. Substituting 3x - 2 for y in the second equation yields 5x + 2(3x - 2) = 40. Distributing gives 5x + 6x - 4 = 40. Combining like terms yields 11x - 4 = 40. Adding 4 to both sides yields 11x = 44. Dividing both sides by 11 yields x = 4.
Choice A is incorrect and may result from conceptual errors.
Choice C is incorrect and may result from an arithmetic error.
Choice D is incorrect and may result from dividing 40 by 5.
Linear Equations in One Variable
Solving equations of the form ax + b = c, including equations with variables on both sides, fractions, and no-solution or infinite-solution cases.
Linear Equations in Two Variables
Working with equations like y = mx + b: interpreting slope and intercepts, converting between forms, and connecting equations to their graphs.
Linear Functions
Modeling real situations with linear functions — constant rates of change, initial values, and evaluating or interpreting f(x) in context.
Linear Inequalities
Solving and interpreting inequalities in one or two variables, including flipping the inequality sign and identifying solution regions.
59 Systems of Linear Equations questions with step-by-step explanations, woven into a day-by-day study plan built for your test date.
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