David made an error in his elimination step. Let’s identify and explain the correct approach to solving the system of equations [tex]\(4x + 6y = 6\)[/tex] and [tex]\(2x + 5y = 11\)[/tex].
Correct Method for Solving the System:
1. Start with the given system of equations:
[tex]\[
\begin{cases}
4x + 6y = 6 & \text{(1)} \\
2x + 5y = 11 & \text{(2)}
\end{cases}
\][/tex]
2. Align the coefficients of [tex]\(x\)[/tex] by multiplying the second equation by 2:
[tex]\[
2 \cdot (2x + 5y) = 2 \cdot 11
\][/tex]
This transforms the second equation into:
[tex]\[
4x + 10y = 22 \quad \text{(3)}
\][/tex]
3. Now we have the following system of equations:
[tex]\[
\begin{cases}
4x + 6y = 6 & \text{(1)} \\
4x + 10y = 22 & \text{(3)}
\end{cases}
\][/tex]
4. Subtract equation (1) from equation (3) to eliminate [tex]\(x\)[/tex]:
[tex]\[
(4x + 10y) - (4x + 6y) = 22 - 6
\][/tex]
Simplifying this, we get:
[tex]\[
4y = 16 \quad \Rightarrow \quad y = 4
\][/tex]
5. Substitute [tex]\(y = 4\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]. Let's use equation (2):
[tex]\[
2x + 5(4) = 11
\][/tex]
Simplifying the equation, we get:
[tex]\[
2x + 20 = 11 \\
2x = 11 - 20 \\
2x = -9 \\
x = \frac{-9}{2} \\
x = -4.5
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -4.5, \quad y = 4
\][/tex]
Explanation of David’s Error:
David's error occurred when he subtracted the equations incorrectly. Here's where he went wrong:
- After multiplying the second equation by 2, he got [tex]\(4x + 10y = 22\)[/tex], which is correct.
- He then set up the correct subtraction: [tex]\( (4x + 6y) - (4x + 10y) \)[/tex] but simplified it incorrectly.
- The correct subtraction should result in: [tex]\(10y - 6y\)[/tex], not [tex]\(5y\)[/tex].
- Consequently, his error resulted in an incorrect value for [tex]\(y\)[/tex] and subsequently incorrect results for [tex]\(x\)[/tex].
Correct subtraction of the equations and solving them as shown yields the solution [tex]\(x = -4.5\)[/tex] and [tex]\(y = 4\)[/tex].
