Let's match each linear equation with its corresponding graph based on the image.

We have five equations and five graphs, color-coded as follows:

1. $y = 5x$
2. $y = -2$
3. $y = -3x$
4. $y = -\frac{1}{2}x + 1$
5. $y = -5x + 1$

Step-by-Step Matching:

• Red (R): The red line has a steep negative slope and intersects the y-axis at $y = 1$. This matches equation 5: $y = -5x + 1$.

• Green (G): The green line has a positive slope, passing through the origin. This corresponds to equation 1: $y = 5x$.

• Purple (P): The purple line has a negative slope, with the steepness between the red and blue lines. This matches equation 3: $y = -3x$.

• Black (K): The black line is horizontal, so the slope is zero. It intersects the y-axis at $y = -2$. This matches equation 2: $y = -2$.

• Blue (B): The blue line has a gentle negative slope and intersects the y-axis at $y = 1$. This corresponds to equation 4: $y = -\frac{1}{2}x + 1$.

Final Matching:

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

Would you like more details or have any questions about this solution?

Here are some follow-up questions for deeper understanding:

1. What is the relationship between the slope and steepness of a line?
2. How does changing the y-intercept affect the position of the line?
3. What does the negative sign in front of the slope indicate for the line's direction?
4. Why is the black line horizontal, and what is its slope?
5. How would the graph of $y = 5x - 3$ differ from $y = 5x$?

Tip:

For any linear equation $y = mx + b$, the slope $m$ determines the steepness and direction, while $b$ is the y-intercept, showing where the line crosses the y-axis.