Question

Solve the following system of equations graphically on the set of axes below.
y, equals, one quarter, x, plus, 4
y=
4
1
​







x+4
3, x, plus, 2, y, equals, minus, 6
3x+2y=−6

Answer 1

To solve the given system of equations graphically, we first need to identify both equations clearly.

The given equations are:

1. \( y = \frac{1}{4}x + 4 \) (Equation 1)
2. \( 3x + 2y = -6 \) (Equation 2)

We'll convert Equation 2 into slope-intercept form (\( y = mx + b \)) to make it easier to graph:

Starting from \( 3x + 2y = -6 \):
1. Subtract \( 3x \) from both sides:
\[
2y = -3x - 6
\]
2. Divide each term by 2:
\[
y = -\frac{3}{2}x - 3 \quad \text{(Equation 2 in slope-intercept form)}
\]

Now we have both equations in slope-intercept form:
1. \( y = \frac{1}{4}x + 4 \)
2. \( y = -\frac{3}{2}x - 3 \)

### Step 1: Graph the Equations

#### For Equation 1: \( y = \frac{1}{4}x + 4 \)
- **Y-intercept (b)**: 4 (point is (0, 4))
- **Slope (m)**: \( \frac{1}{4} \) (For every 4 units you move right, move up 1 unit)
- **Another point**:
1. When \( x = 4 \): \( y = \frac{1}{4}(4) + 4 = 5 \) (point is (4, 5))

Plot the points (0, 4) and (4, 5) and draw the line through them.

#### For Equation 2: \( y = -\frac{3}{2}x - 3 \)
- **Y-intercept (b)**: -3 (point is (0, -3))
- **Slope (m)**: \( -\frac{3}{2} \) (For every 2 units you move right, move down 3 units)
- **Another point**:
1. When \( x = 2 \): \( y = -\frac{3}{2}(2) - 3 = -6 \) (point is (2, -6))

Plot the points (0, -3) and (2, -6) and draw the line through them.

### Step 2: Finding the Intersection
After plotting both lines, look for the point where they intersect, as this point represents the solution to the system of equations.

### Step 3: Determining the Intersection Point
To determine the intersection point algebraically or estimate it on the graph, set the two equations equal to each other:

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

To solve for \( x \):

1. Multiply the entire equation by 4 to eliminate the fraction:
\[
x + 16 = -6x - 12
\]
2. Combine like terms:
\[
x + 6x = -12 - 16
\]
\[
7x = -28
\]
3. Divide by 7:
\[
x = -4
\]

Now substitute \( x = -4 \) back into one of the original equations to solve for \( y \):
Using \( y = \frac{1}{4}(-4) + 4 \):
\[
y = -1 + 4 = 3
\]

### Solution
The intersection point, which is the solution to the system of equations, is \((-4, 3)\).

You can now mark this point on your graph where the two lines intersect.