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Which system of linear equations can be solved using equation [x y] = [1/4,3/4,1,2] [28,-12]

Which system of linear equations can be solved using equation [x y] = [1/4,3/4,1,2] [28,-12]-example-1
User Ralph King
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2 Answers

1 vote

The correct option is a.

The correct system of linear equations that can be solved using the given equation is:


$-8x + 3y = 28$


$4x - y = -12$

To determine which system of linear equations can be solved using the given equation
$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right]$, let's perform matrix multiplication to find the values of x and y:


$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right]$

Step 1: Multiply the matrices.


$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right] = \left[\begin{array}{c}(1)/(4) \cdot 28 + (3)/(4) \cdot (-12) \\ 1 \cdot 28 + 2 \cdot (-12)\end{array}\right]$

Step 2: Calculate the values.


$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}7 - 9 \\ 28 - 24\end{array}\right] = \left[\begin{array}{c}-2 \\ 4\end{array}\right]$

So, the values of x and y are
$x = -2$ and
$y = 4$.

Now, let's check which system of equations corresponds to this solution.

System 1:


$(1)/(4) x + y = 28$


$(3)/(4) x + 2y = -12$

Plugging in the values we found:


$(1)/(4) (-2) + 4 = -0.5 + 4 = 3.5 \\eq 28$ (Not true)


$(3)/(4) (-2) + 2(4) = -1.5 + 8 = 6.5 \\eq -12$ (Not true)

System 2:


$-8x + 3y = 28$


$4x - y = -12$

Plugging in the values we found:


$-8(-2) + 3(4) = 16 + 12 = 28$ (True)


$4(-2) - 4 = -8 - 4 = -12$ (True)

User Insomaniacal
by
9.1k points
4 votes

The question is illustrated in matrix format.

To solve it we would need to proof that the value of x and y in the matrix corresponds with the answer

User Drs
by
7.5k points

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