Question

or
Solve this system of equations by graphing. First graph the equations, and then type the solution.
y=
1
3
x+4
y=
–
5
6
x–3

Answer 1

To solve the system of equations by graphing, we first rewrite the equations in a more usable format for graphing.

1. **Equation 1**:
\[
y = \frac{1}{3}x + 4
\]
This equation has a slope of \(\frac{1}{3}\) and a y-intercept of 4. To graph this:
- Start at (0, 4) on the y-axis.
- Use the slope \(\frac{1}{3}\) to find another point. From (0, 4), go up 1 unit and right 3 units to the point (3, 5).
- Draw the line through these points.

2. **Equation 2**:
\[
y = -\frac{5}{6}x - 3
\]
This equation has a slope of \(-\frac{5}{6}\) and a y-intercept of -3. To graph this:
- Start at (0, -3) on the y-axis.
- Use the slope \(-\frac{5}{6}\) to find another point. From (0, -3), go down 5 units and right 6 units to the point (6, -8).
- Draw the line through these points.

### Finding the Intersection
After graphing both lines, look for the point where they intersect. The coordinates of this intersection point will provide the solution to the system of equations.

**Intersection Point (Solution)**:
Let's find the intersection algebraically to confirm. We set the equations equal to each other:

\[
\frac{1}{3}x + 4 = -\frac{5}{6}x - 3
\]

Multiply everything by 6 to eliminate fractions:

\[
2x + 24 = -5x - 18
\]

Combine like terms:

\[
2x + 5x = -18 - 24 \\
7x = -42 \\
x = -6
\]

Now, substitute \(x = -6\) back into either original equation to find \(y\):

Using the first equation:

\[
y = \frac{1}{3}(-6) + 4 = -2 + 4 = 2
\]

### Final Solution
The solution is \((-6, 2)\). Therefore, the solution to the system of equations is:

\[
\boxed{(-6, 2)}
\]