Question

How do I eliminate the x-value when solving a system of linear equations?

Answer 1

To eliminate the x-value when solving a system of linear equations, you can use the method of elimination, which involves manipulating the equations so that adding or subtracting them cancels out the variable \( x \). Here’s a step-by-step approach:

### Step-by-Step Guide to Elimination

1. **Write Down the Equations**: Start with your system of equations. For example:
\[
\text{Equation 1: } 2x + 3y = 6
\]
\[
\text{Equation 2: } 4x - y = 5
\]

2. **Multiply to Align Coefficients**: If necessary, multiply one or both equations by a number so that the coefficients of \( x \) are the same (or opposites). For example, if we want the coefficients of \( x \) to be the same:
- Multiply Equation 1 by 2:
\[
4x + 6y = 12 \quad \text{(this is now our new Equation 1)}
\]
- Keep Equation 2 as it is:
\[
4x - y = 5
\]

3. **Subtract or Add the Equations**: Now subtract one equation from the other to eliminate \( x \). In our case:
\[
(4x + 6y) - (4x - y) = 12 - 5
\]
This simplifies to:
\[
6y + y = 7 \quad \Rightarrow \quad 7y = 7
\]
Thus, \( y = 1 \).

4. **Substitute Back to Find \( x \)**: Now that you have \( y \), substitute it back into one of the original equations to find \( x \). Using Equation 1:
\[
2x + 3(1) = 6
\]
This simplifies to:
\[
2x + 3 = 6 \quad \Rightarrow \quad 2x = 3 \quad \Rightarrow \quad x = 1.5
\]

5. **Solution**: The solution to the system of equations is \( x = 1.5 \) and \( y = 1 \).

### Recap
By using the elimination method, you can eliminate one variable (in this case \( x \)) by aligning the coefficients and either adding or subtracting the equations. Then, substitute back to find the remaining variable.