Question

problem 8 ali and sid graphed this system
$y = -\frac{1}{4}x - 1$
$y = \frac{1}{4}x - 3$
what is the solution to the system of equations?

Explanation:

Step1: Set the two equations equal

To find the solution of the system \(

$$\begin{cases}y = -\frac{1}{4}x - 1\\y=\frac{1}{4}x - 3\end{cases}$$

\), we set \(-\frac{1}{4}x - 1=\frac{1}{4}x - 3\) because the solution of a system of linear equations is the point where the two lines intersect, so their \(y\)-values are equal at that \(x\)-value.

Step2: Solve for \(x\)

First, add \(\frac{1}{4}x\) to both sides of the equation:
\(-\frac{1}{4}x+\frac{1}{4}x - 1=\frac{1}{4}x+\frac{1}{4}x - 3\)
Simplifying, we get \(- 1=\frac{2}{4}x - 3\), or \(-1=\frac{1}{2}x - 3\).
Then, add 3 to both sides:
\(-1 + 3=\frac{1}{2}x-3 + 3\)
\(2=\frac{1}{2}x\)
Multiply both sides by 2:
\(x = 2\times2=4\)

Step3: Find \(y\) by substituting \(x = 4\) into one of the equations

Let's substitute \(x = 4\) into \(y=\frac{1}{4}x - 3\):
\(y=\frac{1}{4}\times4-3\)
\(y = 1 - 3=-2\)

Answer:

The solution to the system of equations is \((4,-2)\)