# How to Use Elimination to Solve a System of Equations: Word Problems

When confronted with a system of equations, numerous methods can guide you toward the solution, each with its unique process and appeal. One such method, termed ‘Elimination,’ provides a structured, systematic approach. This article will detail the elimination method step-by-step, contextualizing it within word problems.

## A Step-by-step Guide to Using Elimination to Solve a System of Equations: Word Problems

Elimination, as the name suggests, focuses on eliminating one variable to simplify a system of equations. This method is particularly effective for linear equation systems where addition or subtraction can effectively reduce the system to one equation with one unknown.

### Step 1: Decipher the Word Problem

The first step involves **translating the narrative problem into mathematical terms**. Identify the unknown variables and define them. For example, a problem may involve the cost of apples and bananas. Here, the unknowns are the cost of an apple and a banana.

### Step 2: Develop the System of Equations

With the variables identified, the next task is to **formulate a system of equations** based on the problem’s conditions. Suppose the problem states that three apples and two bananas cost \($13\), and five apples and four bananas cost \($23\), you would get two equations: \(3a+2b=13\) and \(5a+4b=23\).

### Step 3: Arrange for Elimination

Before you can eliminate a variable, the equations must be **arranged appropriately**. You should arrange the equations so that one of the variable’s coefficients in one equation is the opposite of the same variable’s coefficient in the other equation. If such a condition doesn’t exist, you might have to multiply one or both equations by a suitable number.

### Step 4: Execute the Elimination

With the equations arranged, it’s time to **add or subtract them to eliminate one variable**. In our example, if we multiply the first equation by \(2\), we obtain: \(6a+4b=26\) and \(5a+4b=23\). Subtraction of the first from the second equation eliminates \(b\), giving \(-a=-3\).

### Step 5: Solve for the Remaining Variable

With one variable eliminated, you’re left with a simple equation to solve. In our case, solving \(-a=-3\) gives the price of an apple.

### Step 6: Substitute and Solve for the Second Variable

Substitute the found value into any original equation to **solve for the second variable**. If the price of an apple is found to be \(3\), substituting this into \(3a+2b=13\) will yield the price of a banana.

### Step 7: Verify Your Solution

Always ensure to **check the solution** by substituting the values back into the original equations. If both equations hold true, you’ve successfully solved the system.

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