Question

1. slope (m) =

y-intercept (b) =
equation:

1. slope (m) =

y-intercept (b) =
equation:

1. slope (m) =

y-intercept (b) =
equation:
find the slope of the line through the pair of points.

1. (5, 0) and (-4, 2)
2. (6, -6) and (6, 2)
3. (12, 3) and (-7, -5)

Explanation:

Problem 16

Step 1: Find slope (m)

Choose two points on the line, e.g., \((-1, 1)\) and \((1, -5)\). Slope formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
\(m=\frac{-5 - 1}{1 - (-1)}=\frac{-6}{2}=-3\)

Step 2: Find y-intercept (b)

The line crosses the y-axis at \((0, -2)\), so \(b = -2\)

Step 3: Write the equation

Using \(y = mx + b\), substitute \(m = -3\) and \(b = -2\)
\(y=-3x - 2\)

Step 1: Find slope (m)

Choose two points, e.g., \((0, 0)\) and \((5, -5)\). Slope formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
\(m=\frac{-5 - 0}{5 - 0}=\frac{-5}{5}=-1\) (Wait, correction: Let's take \((-2, 4)\) and \((3, -5)\). \(m=\frac{-5 - 4}{3 - (-2)}=\frac{-9}{5}\)? No, better points: \((0,0)\) and \((1, -2)\)? Wait, looking at the graph, the line passes through \((0,0)\) and \((5, -10)\)? Wait, no, let's re-examine. Wait, the line in problem 17: let's take two clear points. Let's say \((-3, 5)\) and \((2, -5)\). Then \(m=\frac{-5 - 5}{2 - (-3)}=\frac{-10}{5}=-2\). Wait, maybe I made a mistake earlier. Let's do it properly. Let's take \((0,0)\) and \((5, -10)\)? No, the graph: when x=0, y=0; when x=1, y=-2? Wait, no, let's count the rise over run. From (0,0) to (5, -10)? No, the line goes from ( - 2, 4) to (3, -6)? Wait, maybe the correct slope is -2. Wait, let's use two points: (0,0) and (1, -2)? No, let's check the y-intercept. Wait, the line passes through the origin, so \(b = 0\). Let's take (0,0) and (5, -10)? No, maybe ( - 1, 2) and (2, -4). Then \(m=\frac{-4 - 2}{2 - (-1)}=\frac{-6}{3}=-2\). Yes, that makes sense. So slope \(m = -2\), y-intercept \(b = 0\) (since it passes through (0,0)). Then equation \(y = -2x + 0\) or \(y = -2x\). Wait, let's verify with another point. If x=1, y=-2. Looking at the graph, that seems correct.

Step 1 (corrected): Find slope (m)

Take points \((-1, 2)\) and \((2, -4)\). Slope \(m=\frac{-4 - 2}{2 - (-1)}=\frac{-6}{3}=-2\)

Step 2: Find y-intercept (b)

The line crosses the y-axis at (0,0), so \(b = 0\)

Step 3: Write the equation

\(y = -2x + 0\) or \(y = -2x\)
Wait, maybe my initial point selection was wrong. Let's re-express. Let's take (0,0) and (1, -2). Then \(m=\frac{-2 - 0}{1 - 0}=-2\), correct. So slope is -2, y-intercept 0, equation \(y = -2x\).

Step 1: Find slope (m)

Choose two points, e.g., \((0, -1)\) and \((5, 0)\). Slope formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
\(m=\frac{0 - (-1)}{5 - 0}=\frac{1}{5}\)? Wait, no, let's take ( - 5, -2) and (5, 0). Then \(m=\frac{0 - (-2)}{5 - (-5)}=\frac{2}{10}=\frac{1}{5}\). Wait, the line is rising from left to right, so positive slope. Let's count rise over run. From ( - 5, -2) to (5, 0): run is 10, rise is 2, so slope \(m=\frac{2}{10}=\frac{1}{5}\)? Wait, no, maybe (0, -1) and (5, 0): run 5, rise 1, so slope \(\frac{1}{5}\). Wait, but maybe better points: ( - 3, -1.6) and (2, -0.6)? No, let's do it properly. Let's take (0, -1) as the y-intercept (since it crosses the y-axis at (0, -1)? Wait, no, the graph: when x=0, y=-1? Wait, no, looking at the graph, the line passes through (0, -1) and (5, 0)? Wait, no, the line in problem 18: let's see, when x= -5, y= -2; x=0, y= -1; x=5, y=0. So the slope is \(\frac{0 - (-1)}{5 - 0}=\frac{1}{5}\)? Wait, no, from ( - 5, -2) to (0, -1): rise is 1, run is 5, so slope \(\frac{1}{5}\). Then y-intercept is -1 (since it crosses the y-axis at (0, -1)). Then equation \(y=\frac{1}{5}x - 1\). Wait, but let's check with (5, 0): \(y=\frac{1}{5}(5) - 1 = 1 - 1 = 0\), correct. So:

Step 1: Find slope (m)

Using points \((-5, -2)\) and \((0, -1)\): \(m=\frac{-1 - (-2)}{0 - (-5)}=\frac{1}{5}\)

Step 2: Find y-intercept (b)

The line crosses the y-axis at \((0, -1)\), so \(b = -1\)

Step 3: Write the equation

\(y=\frac{1}{5}x - 1\)

Answer:

slope (m) = \(-3\)
y-intercept (b) = \(-2\)
equation: \(y = -3x - 2\)

Problem 17