Math Problem Statement

Determine the equations of the three given lines based on their graphs.

Solution

To find the equations of the lines shown in each graph, we can use the slope-intercept form of a line, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

First graph:

  1. Y-intercept: The line crosses the y-axis at y=2y = -2, so b=2b = -2.
  2. Slope: The slope is calculated as riserun\frac{\text{rise}}{\text{run}}. From one point (e.g., (0,2)(0, -2)) to another (e.g., (2,3)(2, -3)), the rise is 3(2)=1-3 - (-2) = -1 and the run is 20=22 - 0 = 2. Therefore, the slope is m=12m = \frac{-1}{2}.
    • Equation: y=12x2y = -\frac{1}{2}x - 2.

Second graph:

  1. Y-intercept: The line crosses the y-axis at y=1y = -1, so b=1b = -1.
  2. Slope: Using the points (0,1)(0, -1) and (3,3)(3, -3), the rise is 3(1)=2-3 - (-1) = -2 and the run is 30=33 - 0 = 3, so the slope is m=23m = \frac{-2}{3}.
    • Equation: y=23x1y = -\frac{2}{3}x - 1.

Third graph:

  1. Y-intercept: The line crosses the y-axis at y=2y = 2, so b=2b = 2.
  2. Slope: Using the points (0,2)(0, 2) and (3,3)(3, 3), the rise is 32=13 - 2 = 1 and the run is 30=33 - 0 = 3, so the slope is m=13m = \frac{1}{3}.
    • Equation: y=13x+2y = \frac{1}{3}x + 2.

Would you like a more detailed explanation or have any other questions?

Here are some related questions you might explore:

  1. What happens when the slope is zero?
  2. How does changing the y-intercept affect the graph of the line?
  3. How can you derive the slope if only two points are given?
  4. Can you determine the slope if the line is vertical or horizontal?
  5. How would you graph a line given its equation in point-slope form?

Tip: Remember, the slope describes the steepness of a line. A negative slope means the line is falling, while a positive slope means it's rising.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Graphing

Formulas

Slope-intercept form: y = mx + b
Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Slope-intercept form of a line

Suitable Grade Level

Grades 8-10