Solving Systems Using Elimination Method

Grade 8 Mathematics Common Core State Standards

What is a System of Equations?

Two or more equations with the same variables Example: 2x + 3y = 8 and x - y = 1 Solution is the point where lines intersect Must satisfy ALL equations simultaneously

Think About It

If you have 2x + y = 7 and x + y = 4, what values of x and y would make BOTH equations true?

The Elimination Method

Goal: Eliminate one variable by adding or subtracting equations Make coefficients of one variable opposites Add equations to cancel out that variable Solve for the remaining variable Substitute back to find the other variable

Step-by-Step Elimination Process

Example 1: Ready-to-Eliminate

Solve: 3x + 2y = 16 and 3x - 2y = 8 Notice: +2y and -2y are already opposites! Add the equations: (3x + 2y) + (3x - 2y) = 16 + 8 Simplify: 6x = 24, so x = 4 Substitute x = 4 into first equation: 3(4) + 2y = 16 Solve: 12 + 2y = 16, so 2y = 4, so y = 2 Solution: (4, 2)

Your Turn: Practice Problem

Solve this system using elimination: 5x + 3y = 19 5x - 3y = 1 Work with a partner and show all steps!

Example 2: Need to Prepare First

Solve: 2x + 3y = 7 and x + y = 3 Coefficients aren't opposites yet Multiply second equation by -2: -2x - 2y = -6 Now we have: 2x + 3y = 7 and -2x - 2y = -6 Add equations: (2x + 3y) + (-2x - 2y) = 7 + (-6) Simplify: y = 1 Substitute back: x + 1 = 3, so x = 2 Solution: (2, 1)

When to Use Elimination vs Substitution

{"left":"Use Elimination When:\nCoefficients are the same or opposites\nBoth equations are in standard form\nVariables have 'nice' coefficients","right":"Use Substitution When:\nOne variable is already isolated\nCoefficients are messy fractions\nOne equation is much simpler"}

Key Takeaways

Elimination method removes one variable temporarily Make coefficients opposites, then add equations Always check your solution in both original equations Choose the most efficient method for each problem Practice makes perfect - try different types of systems!