To solve the system of equations graphically, you will need to plot the two lines represented by the equations:
- Equation 1: \( y = 3x - 8 \)
- Equation 2: \( y = -\frac{1}{3}x + 2 \)
Step 1: Plot the First Line
Equation: \( y = 3x - 8 \)
- Intercept: The y-intercept is \( -8 \) (point \( (0, -8) \)).
- Slope: The slope is \( 3 \), meaning that for every 1 unit increase in \( x \), \( y \) increases by 3 units.
You can find another point using \( x = 1 \):
- When \( x = 1 \):
\( y = 3(1) - 8 = -5 \) (point \( (1, -5) \))
Another point using \( x = 2 \):
- When \( x = 2 \):
\( y = 3(2) - 8 = -2 \) (point \( (2, -2) \))
Step 2: Plot the Second Line
Equation: \( y = -\frac{1}{3}x + 2 \)
- Intercept: The y-intercept is \( 2 \) (point \( (0, 2) \)).
- Slope: The slope is \( -\frac{1}{3} \), meaning that for every 1 unit increase in \( x \), \( y \) decreases by \( \frac{1}{3} \).
You can find another point using \( x = 3 \):
- When \( x = 3 \):
\( y = -\frac{1}{3}(3) + 2 = 1 \) (point \( (3, 1) \))
In summary, the key points to plot are:
- For \( y = 3x - 8 \): \( (0, -8) \), \( (1, -5) \), \( (2, -2) \)
- For \( y = -\frac{1}{3}x + 2 \): \( (0, 2) \), \( (3, 1) \)
Step 3: Draw the Lines
- Plot the points on a set of axes and connect them to form two lines:
- The line for \( y = 3x - 8 \) will slope upwards.
- The line for \( y = -\frac{1}{3}x + 2 \) will slope downwards.
Step 4: Find the Intersection
The solution to the system of equations is where the two lines intersect. You can observe the graph to find this intersection point (if plotted accurately). You could approximate or compute the exact intersection algebraically if needed.
Final Note
As I cannot physically plot the lines for you, you would complete the graphical solution by using graph paper, a graphing calculator, or software to visualize the functions and find their intersection point.
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