systems of equations elimination worksheet

3 min read 23-11-2024

systems of equations elimination worksheet

Meta Description: Conquer systems of equations! This guide provides a step-by-step approach to solving systems of equations using the elimination method, complete with practice problems and solutions. Master this essential algebra skill with our comprehensive worksheet and explanations.

Introduction to Solving Systems of Equations by Elimination

A system of equations is a set of two or more equations with the same variables. Solving a system means finding the values of the variables that satisfy all equations simultaneously. One common method for solving systems of equations is the elimination method (also known as the addition method). This method involves manipulating the equations to eliminate one variable, allowing you to solve for the other. This guide will walk you through the process with examples and a practice worksheet.

Understanding the Elimination Method

The core idea behind elimination is to add or subtract the equations to cancel out one variable. This usually requires multiplying one or both equations by a constant to make the coefficients of one variable opposites. Let's break down the steps:

Step 1: Choose a Variable to Eliminate

Look at the coefficients of x and y in both equations. Identify the variable that will be easiest to eliminate. This often involves finding variables with coefficients that are opposites (like 2 and -2) or easily made into opposites (like 3 and -6).

Step 2: Adjust the Equations (if necessary)

If the coefficients aren't opposites, multiply one or both equations by a constant to make them opposites. Remember to multiply every term in the equation by that constant.

Step 3: Add or Subtract the Equations

Add the equations together if the coefficients are opposites. Subtract the equations if the coefficients have the same sign but different magnitudes. The goal is for one variable to cancel out.

Step 4: Solve for the Remaining Variable

Once one variable is eliminated, you'll have a single equation with one variable. Solve for that variable.

Step 5: Substitute and Solve for the Other Variable

Substitute the value you found in step 4 into either of the original equations. Solve for the remaining variable.

Step 6: Check Your Solution

Substitute both values (x and y) into both original equations to verify that they satisfy both equations.

Example: Solving a System of Equations by Elimination

Let's solve the following system using the elimination method:

2x + y = 7
x - y = 2

1. Choose a Variable: Notice that the coefficients of 'y' are opposites (+1 and -1). We'll eliminate 'y'.

2. Adjust Equations: No adjustments are needed in this case.

3. Add the Equations:

(2x + y) + (x - y) = 7 + 2 3x = 9

4. Solve for x:

x = 3

5. Substitute and Solve for y:

Substitute x = 3 into either original equation. Let's use the first one:

2(3) + y = 7 6 + y = 7 y = 1

6. Check the Solution:

Substitute x = 3 and y = 1 into both original equations:

2(3) + 1 = 7 (True) 3 - 1 = 2 (True)

The solution to the system is x = 3 and y = 1.

How to Deal with No Solution or Infinite Solutions

Sometimes, when you solve a system of equations using elimination, you might encounter:

No Solution: If you end up with a false statement (like 0 = 5), the system has no solution. The lines represented by the equations are parallel and never intersect.
Infinite Solutions: If you end up with a true statement (like 0 = 0), the system has infinitely many solutions. The lines represented by the equations are identical, overlapping completely.