To verify Aaron's calculations and identify his mistake, let's break down the problem step-by-step:
We are given the vectors:
[tex]\[ r = \langle 5, -1 \rangle, \][/tex]
[tex]\[ s = \langle 6, 0 \rangle, \][/tex]
[tex]\[ t = \langle -1, -3 \rangle. \][/tex]
We need to perform the following operations: \( 2r + 4s - 7t \).
### Step 1: Scale each vector
First, we scale each vector by their respective coefficients:
[tex]\[
2r = 2 \times \langle 5, -1 \rangle = \langle 10, -2 \rangle,
\][/tex]
[tex]\[
4s = 4 \times \langle 6, 0 \rangle = \langle 24, 0 \rangle,
\][/tex]
[tex]\[
7t = 7 \times \langle -1, -3 \rangle = \langle -7, -21 \rangle.
\][/tex]
### Step 2: Add and subtract the scaled vectors
Next, we add the scaled vectors \( 2r \) and \( 4s \):
[tex]\[
\langle 10, -2 \rangle + \langle 24, 0 \rangle = \langle 10 + 24, -2 + 0 \rangle = \langle 34, -2 \rangle.
\][/tex]
Then, we subtract the vector \( 7t \) from the result:
[tex]\[
\langle 34, -2 \rangle - \langle -7, -21 \rangle = \langle 34 - (-7), -2 - (-21) \rangle = \langle 34 + 7, -2 + 21 \rangle = \langle 41, 19 \rangle.
\][/tex]
### Review of Aaron's Calculation
Aaron determined:
[tex]\[
2r + 4s - 7t = \langle 41, -23 \rangle.
\][/tex]
Comparing this with our correct calculation:
[tex]\[
2r + 4s - 7t = \langle 41, 19 \rangle.
\][/tex]
We can see that Aaron's result for the y-component is incorrect.
### Error Analysis
Aaron made an error in his calculation:
- He added [tex]$\langle 10, -2 \rangle + \langle 24, 0 \rangle + \langle 7, -21 \rangle$[/tex] incorrectly.
- Specifically, he did not correctly distribute the negative sign when subtracting \( 7t \). Instead of subtracting \(\langle -7, -21 \rangle\), he effectively added it, causing the sign error in the y-component.
Therefore, the mistake is that he did not correctly distribute the negative when subtracting [tex]\( 7t \)[/tex].
