To solve the given system of equations graphically, we first need to identify both equations clearly.
The given equations are:
- \( y = \frac{1}{4}x + 4 \) (Equation 1)
- \( 3x + 2y = -6 \) (Equation 2)
We'll convert Equation 2 into slope-intercept form (\( y = mx + b \)) to make it easier to graph:
Starting from \( 3x + 2y = -6 \):
- Subtract \( 3x \) from both sides: \[ 2y = -3x - 6 \]
- 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:
- \( y = \frac{1}{4}x + 4 \)
- \( y = -\frac{3}{2}x - 3 \)
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:
- 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:
- 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 \( x \):
- Multiply the entire equation by 4 to eliminate the fraction: \[ x + 16 = -6x - 12 \]
- Combine like terms: \[ x + 6x = -12 - 16 \] \[ 7x = -28 \]
- 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.
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