Question

Choose a method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &8 x+y=15\\\ &9=2 y+2 x \end{aligned} $$

Short answer

The solution of the system is \(x = 1\) and \(y = 7\).

Step-by-step solution

Multiply the Equations

First, let's multiply the first equation by 2 and the second equation by 4: \[16x + 2y = 30\] and \[8x + 4y = 36\]

Subtract the Equations

Now, subtract the first equation from the second one: \(8x + 4y - 16x - 2y = 36 - 30\). After simplifying we get: \[-8x + 2y = 6\]. Division by -2 leads to: \(4x - y = -3\). Thus, from this equation we find that \(y = 4x + 3\).

Substitute y into the first Equation

The first equation \[8x + y = 15\] becomes: \[8x + 4x + 3 = 15\]. So, solving for x gives \(x = 1\).

Substitute x into the equation for y

Substitute \(x = 1\) into the equation \(y = 4x + 3\): \(y=4*1+3\), so \(y = 7\).