Systems

SAT Math — Systems of Equations ## What Are Systems of Equations? A system of equations is two (or more) equations that share the same variables. You solve for the values of the variables that satisfy ALL equations simultaneously. The SAT tests two main solution methods: substitution and elimination (addition/subtraction). ## Method 1: Substitution Best when one equation already has a variable isolated (or is easy to isolate). Steps: 1. Solve one equation for one variable 2. Substitute that expression into the other equation 3. Solve for the remaining variable 4. Plug back in to find the other variable Example: > y = 2x + 3 > 3x + y = 13 Step 1: y is already isolated (y = 2x + 3) Step 2: Substitute into the second equation: 3x + (2x + 3) = 13 Step 3: 5x + 3 = 13 → 5x = 10 → x = 2 Step 4: y = 2(2) + 3 = 7 Solution: (2, 7) ## Method 2: Elimination Best when neither equation has a variable conveniently isolated, or when the coefficients align for easy cancellation. Steps: 1. Line up the equations 2. Multiply one or both equations so a variable's coefficients are equal (and opposite) 3. Add the equations — the variable cancels out 4. Solve for the remaining variable, then substitute back Example: > 2x + 3y = 12 > 4x − 3y = 6 The y-coefficients are already opposites (+3 and −3). Add the equations: > (2x + 4x) + (3y − 3y) = 12 + 6 > 6x = 18 → x = 3 > Substitute: 2(3) + 3y = 12 → 6 + 3y =…