The image contains a set of linear equations, each paired with a blank graph. I will now explain how to graph each equation and the steps involved.

Equation 29: $y = -\frac{3}{4}x + 2$

• Slope = $-\frac{3}{4}$: The line will decrease by 3 units vertically for every 4 units it moves to the right.
• Y-intercept = 2: The point where the line crosses the y-axis is at (0, 2).
• Steps:
  1. Start at (0, 2).
  2. From (0, 2), move down 3 units and right 4 units to plot another point at (4, -1).
  3. Draw a line through the points.

Equation 30: $y = \frac{2}{5}x + 5$

• Slope = $\frac{2}{5}$: The line increases by 2 units for every 5 units to the right.
• Y-intercept = 5: The point where the line crosses the y-axis is (0, 5).
• Steps:
  1. Start at (0, 5).
  2. Move up 2 units and right 5 units to plot another point at (5, 7).
  3. Draw the line through these points.

Equation 31: $y = -\frac{4}{5}x - 1$

• Slope = $-\frac{4}{5}$: The line decreases by 4 units for every 5 units to the right.
• Y-intercept = -1: The line crosses the y-axis at (0, -1).
• Steps:
  1. Start at (0, -1).
  2. Move down 4 units and right 5 units to plot a point at (5, -5).
  3. Draw the line through these points.

Equation 32: $y = -3x + 1$

• Slope = -3: The line decreases by 3 units for every 1 unit to the right.
• Y-intercept = 1: The line crosses the y-axis at (0, 1).
• Steps:
  1. Start at (0, 1).
  2. Move down 3 units and right 1 unit to plot another point at (1, -2).
  3. Draw the line through these points.

Equation 33: $y = 2x$

• Slope = 2: The line increases by 2 units for every 1 unit to the right.
• Y-intercept = 0: The line passes through the origin (0, 0).
• Steps:
  1. Start at (0, 0).
  2. Move up 2 units and right 1 unit to plot a point at (1, 2).
  3. Draw the line through these points.

Equation 34: $y = 4$

• This is a horizontal line where $y = 4$, meaning the line passes through all points where the y-coordinate is 4.
• Steps:
  1. Plot a horizontal line across the graph at $y = 4$.

Would you like me to assist you with graphing any specific equation or provide further details on any part of this process?

5 related questions:

1. What is the significance of the slope and y-intercept in a linear equation?
2. How does a negative slope affect the direction of the line?
3. Can you explain how to identify whether a line is steep or shallow based on its slope?
4. What happens to the graph of a line if only the y-intercept changes?
5. How do horizontal and vertical lines differ in their equations?

Tip:

Always start by plotting the y-intercept when graphing a linear equation, as it gives you the starting point for drawing the rest of the line.