Question

let \\( \overline { u } = \langle 4,1 \
angle, \overline { v } = \langle 5, - 5 \
angle \\), and \\( \overline { w } = \langle 2,4 \
angle \\). find the vector \\( \overline { x } \\) that satisfies
\\( 10 \overline { u } - \overline { v } + \overline { x } = 9 \overline { x } + \overline { w } \\).
in this case, \\( \overline { x } = \\)

Explanation:

Step1: Rearrange the vector equation

Given \(10\vec{u}-\vec{v}+\vec{x}=9\vec{x}+\vec{w}\), we can rearrange it to solve for \(\vec{x}\). First, move all terms involving \(\vec{x}\) to one side:
\(10\vec{u}-\vec{v}-\vec{w}=9\vec{x}-\vec{x}\)
\(10\vec{u}-\vec{v}-\vec{w}=8\vec{x}\)
Then \(\vec{x}=\frac{1}{8}(10\vec{u}-\vec{v}-\vec{w})\)

Step2: Substitute the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\)

Since \(\vec{u}=\langle4,1
angle\), \(\vec{v}=\langle5, - 5
angle\), and \(\vec{w}=\langle2,4
angle\)

$$
LATEXBLOCK0
$$

\(10\vec{u}-\vec{v}-\vec{w}=\langle40 - 5-2,10 + 5-4
angle=\langle33,11
angle\)

Step3: Calculate \(\vec{x}\)

\(\vec{x}=\frac{1}{8}\langle33,11
angle=\langle\frac{33}{8},\frac{11}{8}
angle\)

Answer:

\(\langle\frac{33}{8},\frac{11}{8}
angle\)