The vectors $$ p, q $$ and $$ r $$ are $$ p\binom{x}{5}, q=\left(\begin{array}{c}x+y \ y\end{array} ight. $$ $$ r=\binom{5}{x} $$ Given that $$ q=r-p $$ find the, yalues of $$ x $$ and $$ y $$

Question

The vectors $$ p, q $$ and $$ r $$ are $$ p\binom{x}{5}, q=\left(\begin{array}{c}x+y \ y\end{array}ight. $$ $$ r=\binom{5}{x} $$ Given that $$ q=r-p $$ find the, yalues of $$ x $$ and $$ y $$

UpStudy AI · Accepted Answer

We are given
$$
p=\begin{pmatrix} x \ 5 \end{pmatrix},\quad q=\begin{pmatrix} x+y \ y \end{pmatrix},\quad r=\begin{pmatrix} 5 \ x \end{pmatrix}.
$$
and the relation
$$
q = r - p.
$$

**Step 1. Write the equation for $$ q = r - p $$:**

Subtract $$ p $$ from $$ r $$:
$$
r - p = \begin{pmatrix} 5 \ x \end{pmatrix} - \begin{pmatrix} x \ 5 \end{pmatrix} = \begin{pmatrix} 5 - x \ x - 5 \end{pmatrix}.
$$
Thus, the equation becomes:
$$
\begin{pmatrix} x+y \ y \end{pmatrix} = \begin{pmatrix} 5-x \ x-5 \end{pmatrix}.
$$

**Step 2. Equate the corresponding components:**

For the first component:
$$
x + y = 5 - x.
$$
For the second component:
$$
y = x - 5.
$$

**Step 3. Solve the system of equations:**

From the second component, we have:
$$
y = x - 5.
$$

Substitute $$ y = x - 5 $$ into the first component:
$$
x + (x - 5) = 5 - x.
$$
Simplify the equation:
$$
2x - 5 = 5 - x.
$$
Add $$ x $$ to both sides:
$$
3x - 5 = 5.
$$
Add $$ 5 $$ to both sides:
$$
3x = 10.
$$
Divide by $$ 3 $$:
$$
x = \frac{10}{3}.
$$

Now substitute $$ x = \frac{10}{3} $$ into $$ y = x - 5 $$:
$$
y = \frac{10}{3} - 5 = \frac{10}{3} - \frac{15}{3} = -\frac{5}{3}.
$$

**Final Answer:**

$$
x = \frac{10}{3},\quad y = -\frac{5}{3}.
$$
$$ x = \frac{10}{3} $$ and $$ y = -\frac{5}{3} $$.

The vectors and are

Given that
find the, yalues of and

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