1. **Problem statement:** We have three lines r, s, and t with given points P and Q. We need to find the slope $m$, y-intercept $b$, and coordinates of points P and Q for each line. Then, match each line with one of the algebraic expressions: $y = -3x + 5$, $y = -\frac{5}{2}x$, $y = \frac{3}{2}x - 1$, and $y = 2 - \frac{x}{2}$. 2. **Formula for slope:** The slope $m$ of a line through points $P(x_1,y_1)$ and $Q(x_2,y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Line r:** Points $P(-2,-2)$ and $Q(2,2)$. Calculate slope: $$m = \frac{2 - (-2)}{2 - (-2)} = \frac{4}{4} = 1$$ Equation form: $y = mx + b$. Use point $P$ to find $b$: $$-2 = 1 \times (-2) + b \Rightarrow -2 = -2 + b \Rightarrow b = 0$$ So line r: $$y = 1x + 0 = x$$ 4. **Line s:** Points $P(-2,2)$ and $Q(2,-2)$. Calculate slope: $$m = \frac{-2 - 2}{2 - (-2)} = \frac{-4}{4} = -1$$ Use point $P$ to find $b$: $$2 = -1 \times (-2) + b \Rightarrow 2 = 2 + b \Rightarrow b = 0$$ So line s: $$y = -1x + 0 = -x$$ 5. **Line t:** Points $P(-2,2)$ and $Q(2,2)$. Calculate slope: $$m = \frac{2 - 2}{2 - (-2)} = \frac{0}{4} = 0$$ Use point $P$ to find $b$: $$2 = 0 \times (-2) + b \Rightarrow b = 2$$ So line t: $$y = 0x + 2 = 2$$ 6. **Match algebraic expressions:** - $y = -3x + 5$ slope $-3$, intercept $5$ (no match) - $y = -\frac{5}{2}x$ slope $-2.5$, intercept $0$ (no match) - $y = \frac{3}{2}x - 1$ slope $1.5$, intercept $-1$ (no match) - $y = 2 - \frac{x}{2} = -\frac{1}{2}x + 2$ slope $-0.5$, intercept $2$ (no match) None of the given expressions exactly match the lines r, s, or t. However, the closest is line t with $y=2$ and the expression $y=2 - \frac{x}{2}$ which has intercept 2 but slope $-0.5$. So the given expressions do not correspond to the lines described by the points. **Final table:** | Line | m | b | P(x1,y1) | Q(x2,y2) | |-------|---|---|----------|----------| | r | 1 | 0 | (-2,-2) | (2,2) | | s | -1| 0 | (-2,2) | (2,-2) | | t | 0 | 2 | (-2,2) | (2,2) |