Question

Solve the linear system. $$ \begin{aligned} &7 x+4 y=22\\\ &-5 x-9 y=15 \end{aligned} $$

Short answer

The solution to the system of equations is \(x = 6, y = -5\).

Step-by-step solution

Multiply the equations to set them up for elimination

The aim is to eliminate one of the variables when we add the two equations together. To accomplish this, we can multiply the first equation by 5 and the second equation by 7. This procedure gives us:\[35x + 20y = 110\] and \[-35x - 63y = 105\]

Add the two new equations to eliminate x

When we add these two equations, the x terms will cancel each other out: \[35x - 35x + 20y - 63y = 110 + 105\] leads to \[-43y = 215\].

Solve for y

Solve the equation \[-43y = 215\] to find the value of y. By dividing both sides by -43, we get: \(y = -5\).

Substitute y back into one of the original equations

We can substitute \(y = -5\) into the first equation to solve for x: \(7x + 4(-5) = 22\). Simplifying leads to \(7x - 20 = 22\) and then to \(7x = 42\).

Solve for x

Finally, we solve the equation \(7x = 42\) to find the value of x by dividing both sides by 7. Thus, \(x = 6\).