Question

how do you solve systems of equations by elimination

Answer 1

Solving systems of equations by elimination is a method used to find the values of variables that satisfy both equations simultaneously. Here are the steps to use the elimination method effectively:

### Steps to Solve by Elimination

1. **Write the System of Equations**: Start with the system of equations you want to solve. For example:
\[
\begin{align*}
2x + 3y &= 16 \quad \text{(1)} \\
4x - 2y &= 8 \quad \text{(2)}
\end{align*}
\]

2. **Align the Equations**: Make sure the equations are aligned with similar variables and constants in a straight line.

3. **Make the Coefficients of One Variable the Same**: Choose one variable to eliminate. You may need to multiply one or both equations by a number to get the coefficients of that variable to match or to be opposites.

For these equations, let’s eliminate \(x\):
- Multiply equation (1) by 2:
\[
4x + 6y = 32 \quad \text{(3)}
\]

4. **Subtract or Add the Equations**: Now, subtract equation (2) from equation (3) to eliminate \(x\):
\[
(4x + 6y) - (4x - 2y) = 32 - 8
\]
Simplifying gives:
\[
8y = 24
\]

5. **Solve for the Remaining Variable**: Divide both sides of the equation by 8:
\[
y = 3
\]

6. **Substitute Back to Find the Other Variable**: Now that you have the value of \(y\), substitute it back into one of the original equations to solve for \(x\). Using equation (1):
\[
2x + 3(3) = 16
\]
\[
2x + 9 = 16
\]
\[
2x = 7
\]
\[
x = \frac{7}{2} \quad \text{or} \quad 3.5
\]

### Summary of Results
The solution to the system of equations is:
\[
x = \frac{7}{2}, \quad y = 3
\]

### Additional Tips
- If both equations can be represented in a simpler form, do it before applying elimination.
- Sometimes, it might be easier to manipulate the equations to get negative coefficients for one variable, allowing that variable to be eliminated directly.
- Always check your solution by substituting both values back into the original equations.