Question

find the solution to the following systems of linear equations by using the elimination method.
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$$\begin{cases}4x + 7y = 22 \\\\ 7x + 4y = 22\\end{cases}$$

Explanation:

Step1: Multiply equations to eliminate

To eliminate a variable, we can multiply the first equation by 7 and the second by 4 to make the coefficients of \(x\) equal.
First equation: \(4x + 7y = 22\) multiplied by 7: \(28x + 49y = 154\)
Second equation: \(7x + 4y = 22\) multiplied by 4: \(28x + 16y = 88\)

Step2: Subtract equations to eliminate \(x\)

Subtract the second new equation from the first:
\((28x + 49y) - (28x + 16y) = 154 - 88\)
Simplify: \(33y = 66\)

Step3: Solve for \(y\)

Divide both sides by 33: \(y = \frac{66}{33} = 2\)

Step4: Substitute \(y\) into original equation

Substitute \(y = 2\) into \(4x + 7y = 22\):
\(4x + 7(2) = 22\)
\(4x + 14 = 22\)
Subtract 14: \(4x = 8\)
Divide by 4: \(x = 2\)

Answer:

\(x = 2\), \(y = 2\)