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Artist 52 [7]
3 years ago
9

-x=-y+2 and 3y+4=2x solve using matrices

Mathematics
1 answer:
MissTica3 years ago
8 0

\begin{cases}-x=-y+2\\3y+4=2x\end{cases}\implies\begin{cases}x-y=-2\\2x-3y=4\end{cases}

In matrix form, this is

\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-2\\4\end{bmatrix}

The coefficient matrix has determinant

\begin{vmatrix}1&-1\\2&-3\end{vmatrix}=-3+2=-1\neq0

so it has an inverse, which is

\begin{bmatrix}1&-1\\2&-3\end{bmatrix}^{-1}=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}

Multiply both sides by the inverse matrix:

\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}-2\\4\end{bmatrix}

\implies\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-10\\-8\end{bmatrix}

so that x=-10 and y=-8.

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2 years ago
Quadrilateral CAMP below is a rhombus. the length PQ is (x+2) units, and the length of QA is (3x-14) units. Which statements bes
valentinak56 [21]

Answer:

<h3>Q cuts the diagonal PA into 2 equal halves, since the diagonals of rhombus meet at right angles.</h3><h3>The value of x is 8.</h3>

Step-by-step explanation:

Given that Quadrilateral CAMP below is a rhombus. the length PQ is (x+2) units, and the length of QA is (3x-14) units

From the given Q is the middle point, which cut the diagonal PA into 2 equal halves.

By the definition of rhombus, diagonals meet at right angles.

Implies that PQ = QA

x+2 = 3x - 14

x-3x=-14-2

-2x=-16

2x = 16

dividing by 2 on both sides, we will get,

x =\frac{16}{2}

<h3>∴ x=8</h3><h3>Since Q cuts the diagonal PA into 2 equal halves, since the diagonals of rhombus meet at right angles we can equate x+2 = 3x-14 to find the value of x.</h3>

The line segment \overrightarrow{PA}=\overrightarrow{PQ}+\overrightarrow{QA}

\overrightarrow{PA}=x+2+3x-14

=4x-12

=4(8)-12 ( since x=8)

=32-12

=20

<h3>∴ \overrightarrow{PA}=20 units</h3>
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3 years ago
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