Question

Find the solution to the system of equations.
You can use the interactive graph below to find the solution.

\[\begin{cases}
8x-4y=16
\\\\
8x+4y=16
\end{cases}\]
\[x= \]
\[y= \]

Answer 1

To solve the system of equations given by:

\[
\begin{cases}
8x - 4y = 16 \quad \text{(1)} \\
8x + 4y = 16 \quad \text{(2)}
\end{cases}
\]

let's first manipulate these equations.

1. From equation (1):
\[
8x - 4y = 16
\]
We can solve for \(y\):
\[
-4y = 16 - 8x
\]
\[
4y = 8x - 16
\]
\[
y = 2x - 4 \quad \text{(3)}
\]

2. From equation (2):
\[
8x + 4y = 16
\]
Similarly, we can solve for \(y\):
\[
4y = 16 - 8x
\]
\[
y = 4 - 2x \quad \text{(4)}
\]

Now we have two expressions for \(y\) from equations (3) and (4):
\[
y = 2x - 4 \quad \text{(3)}
\]
\[
y = 4 - 2x \quad \text{(4)}
\]

Setting these two expressions for \(y\) equal to each other:
\[
2x - 4 = 4 - 2x
\]
Now, solve for \(x\):
\[
2x + 2x = 4 + 4
\]
\[
4x = 8
\]
\[
x = 2
\]

Now substituting \(x = 2\) into either equation (3) or (4) to find \(y\). Let's use equation (3):
\[
y = 2(2) - 4
\]
\[
y = 4 - 4
\]
\[
y = 0
\]

So, the solution to the system of equations is:
\[
x = 2, \quad y = 0
\]

Final answer:
\[ x = 2 \]
\[ y = 0 \]