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On a graph plot y = -3x + 4 + y = 3x - 2

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To plot the equations \( y = -3x + 4 \) and \( y = 3x - 2 \), we first find points for each line and then graph them.

Equation 1: \( y = -3x + 4 \)

  1. Y-intercept: Set \( x = 0 \): \[ y = -3(0) + 4 = 4 \quad \text{(Point: (0, 4))} \]

  2. X-intercept: Set \( y = 0 \): \[ 0 = -3x + 4 \quad \Rightarrow \quad 3x = 4 \quad \Rightarrow \quad x = \frac{4}{3} \quad \text{(Point: (4/3, 0))} \]

  3. Additional point: Set \( x = 1 \): \[ y = -3(1) + 4 = 1 \quad \text{(Point: (1, 1))} \]

Equation 2: \( y = 3x - 2 \)

  1. Y-intercept: Set \( x = 0 \): \[ y = 3(0) - 2 = -2 \quad \text{(Point: (0, -2))} \]

  2. X-intercept: Set \( y = 0 \): \[ 0 = 3x - 2 \quad \Rightarrow \quad 3x = 2 \quad \Rightarrow \quad x = \frac{2}{3} \quad \text{(Point: (2/3, 0))} \]

  3. Additional point: Set \( x = 1 \): \[ y = 3(1) - 2 = 1 \quad \text{(Point: (1, 1))} \]

Plotting the Lines

  1. Plot the points:

    • For \( y = -3x + 4 \): Plot (0, 4), (4/3, 0), and (1, 1).
    • For \( y = 3x - 2 \): Plot (0, -2), (2/3, 0), and (1, 1).

  2. Draw the lines: Connect the points for each line to create straight lines.

Finding the Intersection

Both lines have the point (1, 1) in common. Hence, they intersect at that point.

Graph Summary

  • The line \( y = -3x + 4 \) has a negative slope and crosses the y-axis at (0, 4).
  • The line \( y = 3x - 2 \) has a positive slope and crosses the y-axis at (0, -2).
  • They intersect at (1, 1).

If you plot this on a coordinate grid, you'll have two lines that cross at point (1, 1).