Systems of Equations

• Solve systems of two equations using substitution
• Solve systems of two equations using elimination (addition/subtraction method)
• Interpret systems of equations in word problem contexts
• Determine whether a system has one solution, no solution, or infinitely many solutions
• Finish on one ACT-hard question and say what made it hard

A system of equations has two or more equations with the same variables. To solve by substitution, isolate one variable in one equation and plug it into the other. To solve by elimination, multiply equations so that one variable cancels when you add the equations together. If the system simplifies to a contradiction (like 0 = 5), there is no solution (parallel lines). If it simplifies to an identity (like 0 = 0), there are infinitely many solutions (same line). The ACT may also present systems as word problems: 'Adult tickets cost $8 and child tickets cost $5. If 200 tickets totaling $1,300 were sold, how many were adult tickets?' Harder systems questions hide a parameter. For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, there is no solution when the x and y coefficients are in the same ratio but the constants are not (parallel lines), and infinitely many solutions when all three are in the same ratio (the same line). For example, 2x + 3y = 6 and 4x + ky = 10 have no solution when k = 6, because 2/4 = 3/6 but 6/10 is not 1/2. The hardest questions, at the end of the section: the parameter hides behind a rewrite. When one equation is in slope-intercept form and the other is in standard form, substitute or rewrite them the same way before comparing: y = 2x + 5 and 4x − 2y = −10 look different but are the same line (substituting gives −10 = −10), so the system has infinitely many solutions, while 4x − 2y = 3 gives a line with the same slope and a different intercept, so there is no solution. A statement such as −10 = 3 after substituting is not an arithmetic slip; it means no solution.

• Choose substitution when one variable is already isolated; choose elimination when coefficients line up nicely.
• Always check your solution by plugging back into both original equations.
• On the ACT, if a system problem looks complex, try plugging in the answer choices. It can be faster.