Question

Solve the linear system. $$ \begin{aligned} &8 x+9 y=42\\\ &6 x-y=16 \end{aligned} $$

Short answer

Therefore, the solution to the system is x = 3, y = 2.

Step-by-step solution

Isolate one variable

Let's start by isolating one of the variables in the second equation. Choosing to isolate y would be a good choice here. The equation, \(6x - y = 16\) can be rewritten as \(y = 6x - 16\)

Substitute the isolated variable into the first equation

Next, substitute the value of y in terms of x from second equation into the first equation. This gives us: \(8x + 9(6x - 16) = 42\). This simplifies to \(62x - 144 = 42\), which further simplifies to \(62x = 186\). Following through, we find that \(x = 3\)

Substitute x into the isolated variable

Having found x, We substitute x=3 into the y-equation we intially isolated to find y: \(y = 6 * 3 - 16 = 2\)

Verification

Finally, we substitute \(x = 3\) and \(y = 2\) into the original equations to verify they are correct solutions. \(8*3 + 9*2 = 42\) and \(6*3 - 2 = 16\), which verifies our solution.