To match the linear equations with their corresponding graphs, let's analyze the equations and the graphs based on their slopes and y-intercepts:

1. $y = -3$:

   • This is a horizontal line at $y = -3$ since the slope is 0.
   • Red line (R) matches this equation.

2. $y = x + 2$:

   • This has a slope of 1 and a y-intercept at 2.
   • Green line (G) has a slope of 1 and intercepts the y-axis at $y = 2$.

3. $y = 5x + 2$:

   • This has a steep positive slope of 5 and a y-intercept at 2.
   • Purple line (P) has a steep slope and matches this equation.

4. $y = \frac{1}{2}x - 4$:

   • The slope is $\frac{1}{2}$ and the y-intercept is -4.
   • Blue line (B) has this slope and y-intercept.

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

   • The slope is $\frac{1}{4}$ and the y-intercept is -4.
   • Black line (K) matches this equation, as it has a gentler slope compared to the blue line.

Summary of Matches:

• $y = -3$ → Red (R)
• $y = x + 2$ → Green (G)
• $y = 5x + 2$ → Purple (P)
• $y = \frac{1}{2}x - 4$ → Blue (B)
• $y = \frac{1}{4}x - 4$ → Black (K)

Would you like further details on any of these equations or graphs?

Here are 5 related questions:

1. How do you determine the slope of a line given its equation?
2. What is the significance of the y-intercept in a linear equation?
3. How can we graph a line if we only know its slope and one point?
4. Why is the slope important for distinguishing between different linear graphs?
5. How do parallel lines differ in terms of their equations?

Tip: To graph a line, always start with the y-intercept (where the line crosses the y-axis) and then use the slope to determine its direction and steepness.