The task in the image is to match each set of points with the correct equation of the straight line passing through them.

Let's solve it step by step. For each pair of points, we will calculate the slope, and then use the point-slope form to find the corresponding equation.

Equations of lines:

1. $y = 5x + 3$
2. $y = -\frac{x}{4} - 2$
3. $y = x - 8$
4. $y = -3x + 5$

Point sets:

1. $(9,1)$ and $(5,-3)$
2. $(1,8)$ and $(-2,-7)$
3. $(-2,11)$ and $(3,-4)$
4. $(-4,-1)$ and $(4,-3)$

Steps:

1. For points $(9,1)$ and $(5,-3)$:

• Slope $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{5 - 9} = \frac{-4}{-4} = 1$.
• Equation: $y = mx + b$. Using $(9,1)$, we have $1 = 1(9) + b \Rightarrow b = -8$.
• Equation: $y = x - 8$.
• Matches with equation 3.

2. For points $(1,8)$ and $(-2,-7)$:

• Slope $m = \frac{-7 - 8}{-2 - 1} = \frac{-15}{-3} = 5$.
• Equation: $y = 5x + b$. Using $(1,8)$, we have $8 = 5(1) + b \Rightarrow b = 3$.
• Equation: $y = 5x + 3$.
• Matches with equation 1.

3. For points $(-2,11)$ and $(3,-4)$:

• Slope $m = \frac{-4 - 11}{3 + 2} = \frac{-15}{5} = -3$.
• Equation: $y = -3x + b$. Using $(-2,11)$, we have $11 = -3(-2) + b \Rightarrow b = 5$.
• Equation: $y = -3x + 5$.
• Matches with equation 4.

4. For points $(-4,-1)$ and $(4,-3)$:

• Slope $m = \frac{-3 - (-1)}{4 - (-4)} = \frac{-3 + 1}{4 + 4} = \frac{-2}{8} = -\frac{1}{4}$.
• Equation: $y = -\frac{1}{4}x + b$. Using $(-4,-1)$, we have $-1 = -\frac{1}{4}(-4) + b \Rightarrow b = -2$.
• Equation: $y = -\frac{1}{4}x - 2$.
• Matches with equation 2.

Final Matching:

1. (9, 1), (5, -3) → $y = x - 8$ (Equation 3)
2. (1, 8), (-2, -7) → $y = 5x + 3$ (Equation 1)
3. (-2, 11), (3, -4) → $y = -3x + 5$ (Equation 4)
4. (-4, -1), (4, -3) → $y = -\frac{x}{4} - 2$ (Equation 2)

Would you like further explanation on any of these steps or more details?

5 Related Questions:

1. How do you calculate the slope between two points?
2. What is the point-slope form of a line equation?
3. How can you determine the y-intercept from a given point and slope?
4. How can different forms of a line equation be transformed into each other?
5. What are real-life examples where calculating the equation of a line is useful?

Tip:

Always double-check the signs when calculating the slope, as a small error can change the entire equation.