Objective: Learn to solve linear systems using the addition method.
Goal: Obtain two equations whose sum results in a single-variable equation.
Key Steps
Equation Preparation:
Aim to have coefficients for a variable that differ only in sign.
May require multiplying one or both equations by a non-zero number so that coefficients of one of variables, x or y, become opposites. Then when the two equations are added, this variable is eliminated.
Steps to Solve by Addition Method:
Step 1: Rewrite both equations in the form (ax + by = c), if necessary.
Step 2: If necessary, multiply equations so that x or y coefficients sum to zero.
Step 3: Add the equations to eliminate one variable, resulting in a single-variable equation.
Step 4: Solve the resulting equation in one variable.
Step 5: Substitute back value obtained step four into either of given equations to find the other variable.
Step 6: Verify the solution in original equations.
Example Problem
Given Equations:
(4x = 5 - 3y)
(3x = 2y - 9)
Solution Steps:
Step 1: Convert to standard form: Ax + By = C
(4x + 3y = 5)
(3x - 2y = -9)
Step 2: Adjust equations to eliminate (y): multiply either or both equations by appropriate numbers so that sum of either co-efficient is zero.
Multiply first equation by 2: (8x + 6y = 10)
Multiply second equation by 3: (9x - 6y = -27)
Step 3: Add equations:
Result: (17x = -17)
Step 4: Solve for (x): solve equation in one variable. Dividing both sides of 17x = -17
(x = -1)
Step 5: Substitute (x) back: and find value for the other variable.
Original equation: (3x = 2y - 9)
Substitute: (3(-1) = 2y - 9)
Simplify: (6 = 2y) - adding 9 to both sides - dividing by 2: leading to (y = 3)
Solution: Ordered pair ((-1, 3)).
Verification
Check the solution ((-1, 3)) in both original equations to confirm it satisfies the system.