Math Problem Statement

These two lines are parallel. Write an equation for each.

Solution

The question asks for equations of two parallel lines, labeled aa and bb, using the graph provided. Here is a step-by-step breakdown:

Step 1: Identify the slope (mm)

Parallel lines have the same slope. To find the slope of line aa, pick two clear points on the graph:

  • Point 1: (0,1)(0, -1)
  • Point 2: (2,2)(2, 2)

The slope (mm) is calculated using the formula: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} m=2(1)20=32m = \frac{2 - (-1)}{2 - 0} = \frac{3}{2}

Thus, the slope of line aa is 32\frac{3}{2}, and line bb also has the same slope since the lines are parallel.

Step 2: Find the y-intercepts (bb)

The slope-intercept form of a line is y=mx+by = mx + b. To find bb, substitute the slope and one point on the line.

For line aa: The y-intercept is already visible on the graph, which is b=1b = -1. The equation is: y=32x1y = \frac{3}{2}x - 1

For line bb: From the graph, the y-intercept of line bb is (0,3.2)(0, -3.2), so b=3.2b = -3.2. The equation is: y=32x3.2y = \frac{3}{2}x - 3.2

Final Answer:

  • Line aa: y=32x1y = \frac{3}{2}x - 1
  • Line bb: y=32x3.2y = \frac{3}{2}x - 3.2

Would you like further details or explanations?
Here are some additional questions to expand your understanding:

  1. How do we find the slope of a line from a graph?
  2. What does it mean for two lines to be parallel in terms of their slopes?
  3. How can we verify that two given equations represent parallel lines?
  4. How can the slope-intercept form of a line be converted into the standard form?
  5. What is the significance of the y-intercept in a linear equation?

Tip: Always ensure you pick clear and accurate points from the graph when calculating the slope to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Parallel Lines
Slope-Intercept Form

Formulas

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

Theorems

Parallel lines theorem: Parallel lines have the same slope

Suitable Grade Level

Grades 8-10