Question

On a coordinate plane, a line goes through (0, 4) and (2, 0).
The first equation from the previous system of equations is graphed. Graph the second equation to find the solution of the system of equations.
y = −2x + 4,
y = −
1
3
x − 1
What is the solution to the system?

Answer 1

To find the solution to the system of equations given by:

1. \( y = -2x + 4 \)
2. \( y = -\frac{1}{3}x - 1 \)

Let's first identify the points where the two lines intersect, which is the solution to the system of equations.

### Step 1: Graphing the first equation

The first equation \( y = -2x + 4 \) has a y-intercept of 4 and a slope of -2. Therefore:

- When \( x = 0 \), \( y = 4 \) (point (0, 4))
- When \( x = 2 \), \( y = 0 \) (point (2, 0))

The line can be drawn through these points.

### Step 2: Graphing the second equation

The second equation \( y = -\frac{1}{3}x - 1 \) has a y-intercept of -1 and a slope of -\(\frac{1}{3}\). Therefore:

- When \( x = 0 \), \( y = -1 \) (point (0, -1))
- When \( x = 3 \), \( y = 0 \) (point (3, 0))

You can plot these points and draw the line through them.

### Step 3: Finding the intersection point

To find the intersection point algebraically, set the two equations equal to each other:

\[
-2x + 4 = -\frac{1}{3}x - 1
\]

### Step 4: Solve for \( x \)

To eliminate the fraction, multiply through by 3:

\[
-6x + 12 = -x - 3
\]

Now, add \( 6x \) to both sides:

\[
12 = 5x - 3
\]

Now, add 3 to both sides:

\[
15 = 5x
\]

Divide both sides by 5:

\[
x = 3
\]

### Step 5: Substitute \( x \) back into one of the equations to find \( y \)

Using the value of \( x \) in the first equation:

\[
y = -2(3) + 4 = -6 + 4 = -2
\]

### Conclusion

The solution to the system of equations is:

\[
\boxed{(3, -2)}
\]

You can check by plugging \( x = 3 \) into the second equation:

\[
y = -\frac{1}{3}(3) - 1 = -1 - 1 = -2
\]

Both lines intersect at the point (3, -2), validating our solution.