Question
Given the system of equations below:
3x-2y=10
3x+2y=14
Part I: Using the elimination method, describe the steps you would take to solve the system by eliminating the y
terms. (You are NOT required to solve anything here.)
3x-2y=10
3x+2y=14
Part I: Using the elimination method, describe the steps you would take to solve the system by eliminating the y
terms. (You are NOT required to solve anything here.)
Answers
GPT-4o mini
To solve the system of equations using the elimination method by eliminating the \(y\) terms, follow these steps:
1. **Write down the system of equations**:
\[
\begin{align*}
3x - 2y &= 10 \quad \text{(Equation 1)} \\
3x + 2y &= 14 \quad \text{(Equation 2)}
\end{align*}
\]
2. **Align the equations**:
Make sure both equations are aligned so that \(x\) and \(y\) coefficients are positioned correctly.
3. **Identify the coefficients of \(y\)**:
In Equation 1, the coefficient of \(y\) is \(-2\), and in Equation 2, it is \(+2\).
4. **Multiply one or both equations if necessary**:
To eliminate \(y\), you can add the two equations directly since the coefficients of \(y\) are already opposites (\(-2\) and \(+2\)).
5. **Add the two equations**:
\[
(3x - 2y) + (3x + 2y) = 10 + 14
\]
When you add the left side, the \(y\) terms cancel out:
\[
3x + 3x + (-2y + 2y) = 24
\]
6. **Simplify the resulting equation**:
This simplifies to:
\[
6x = 24
\]
7. **Solve for \(x\)** (though we don't need to solve it, just recognize this step):
At this point, you would isolate \(x\) by dividing both sides by 6, if you were to continue solving.
By following these steps, you successfully eliminate the \(y\) terms, allowing you to solve for \(x\) in the equation derived from the combination of the original equations.
1. **Write down the system of equations**:
\[
\begin{align*}
3x - 2y &= 10 \quad \text{(Equation 1)} \\
3x + 2y &= 14 \quad \text{(Equation 2)}
\end{align*}
\]
2. **Align the equations**:
Make sure both equations are aligned so that \(x\) and \(y\) coefficients are positioned correctly.
3. **Identify the coefficients of \(y\)**:
In Equation 1, the coefficient of \(y\) is \(-2\), and in Equation 2, it is \(+2\).
4. **Multiply one or both equations if necessary**:
To eliminate \(y\), you can add the two equations directly since the coefficients of \(y\) are already opposites (\(-2\) and \(+2\)).
5. **Add the two equations**:
\[
(3x - 2y) + (3x + 2y) = 10 + 14
\]
When you add the left side, the \(y\) terms cancel out:
\[
3x + 3x + (-2y + 2y) = 24
\]
6. **Simplify the resulting equation**:
This simplifies to:
\[
6x = 24
\]
7. **Solve for \(x\)** (though we don't need to solve it, just recognize this step):
At this point, you would isolate \(x\) by dividing both sides by 6, if you were to continue solving.
By following these steps, you successfully eliminate the \(y\) terms, allowing you to solve for \(x\) in the equation derived from the combination of the original equations.