To solve the system of equations by graphing, we'll first graph both equations:
- Equation 1: \( y = \frac{1}{4}x + 4 \)
- Equation 2: \( y = -\frac{3}{2}x - 3 \)
Step 1: Graphing the First Equation \( y = \frac{1}{4}x + 4 \)
- Y-Intercept: When \( x = 0 \), \( y = 4 \). So the point is (0, 4).
- Slope: The slope is \( \frac{1}{4} \), which means for every 4 units you move to the right (increase in \( x \)), you move 1 unit up (increase in \( y \)).
Using the y-intercept and the slope, we can plot another point:
- From (0, 4), move 4 units to the right to (4, 4) and 1 unit up to (4, 5). Now we have two points: (0, 4) and (4, 5).
Draw the line through these points.
Step 2: Graphing the Second Equation \( y = -\frac{3}{2}x - 3 \)
- Y-Intercept: When \( x = 0 \), \( y = -3 \). So the point is (0, -3).
- Slope: The slope is \( -\frac{3}{2} \), meaning for every 2 units you move to the right (increase in \( x \)), you move 3 units down (decrease in \( y \)).
Using the y-intercept:
- From (0, -3), move 2 units to the right to (2, -3) and 3 units down to (2, -6). Now we have two points: (0, -3) and (2, -6).
Draw the line through these points.
Step 3: Finding the Intersection
Once both lines are plotted on the same graph, the intersection point is your solution to the system of equations—the \( (x,y) \) values that satisfy both equations.
Step 4: Finding the Intersection Algebraically
To find the intersection point more accurately, set the two equations equal to one another:
\[ \frac{1}{4}x + 4 = -\frac{3}{2}x - 3 \]
Multiply all terms by 4 to eliminate the fraction:
\[ x + 16 = -6x - 12 \]
Combine like terms:
\[ x + 6x = -12 - 16 \] \[ 7x = -28 \] \[ x = -4 \]
Now substitute \( x = -4 \) into one of the original equations to find \( y \):
Using \( y = \frac{1}{4}(-4) + 4 \):
\[ y = -1 + 4 = 3 \]
So the solution is \( (-4, 3) \).
Conclusion
The graphs will show that the two lines intersect at the point \( (-4, 3) \). This is the solution to the system of equations.
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