Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &y=x-9\\\ &x+8 y=0 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 8\) and \(y = -1\)

Step-by-step solution

Write Equations in Standard Form

Let's start by rewriting the equations in standard form. The standard form is \(Ax + By = C\). The first equation is already in standard form \(y = x - 9\), lets rewrite it as \(x - y = 9\). The second equation is already \(x + 8y = 0\)

Create the Linear Combination

A linear combination involves adding or subtracting equations to eliminate one of the variables. To create a linear combination, there should be the same numeric coefficient in front of x or y in both equations. In this case to make the coefficient of x in both equations the same, we can multiply the first equation by -1. That will give us \(-x + y = -9\) and adding this to the second equation: \(-x + x + y + 8y = -9 + 0\). Simplifying it we have \(9y = -9\)

Solve For y

Now, with one equation and one variable, we can solve for y by dividing both sides of the equation by 9. \(9y / 9 = -9 / 9\), simplifying it we get \(y = -1\)

Substitute y into one of the original equations

Take the value of y and substitute into one of the original equations. Choose \(y = x - 9\), substituting y we get \(-1 = x - 9\). Add 9 to both sides gives \(x = 8\)