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

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Algebra Questions

The vectors and are Given that find the, yalues of and