Math Problem Statement

Use the graph to write an equation to represent the linear relationship in slope-intercept form.

Solution

To find the equation of the line represented in the graph, we use the slope-intercept form of a line:

y=mx+by = mx + b where:

  • mm is the slope of the line.
  • bb is the yy-intercept (the point where the line crosses the yy-axis).

Step 1: Find the slope (mm)

The slope is given by:

m=riserunm = \frac{\text{rise}}{\text{run}}

From the graph:

  • The line passes through the points (8,6)(-8, 6) and (0,2)(0, -2).
  • Calculate the slope between these two points:

m=y2y1x2x1=260(8)=88=1m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 6}{0 - (-8)} = \frac{-8}{8} = -1

Thus, the slope m=1m = -1.


Step 2: Find the yy-intercept (bb)

The yy-intercept is the point where x=0x = 0. From the graph, we observe that:

b=2b = -2


Step 3: Write the equation

Substitute the values of mm and bb into the slope-intercept form:

y=x2y = -x - 2


Final Answer:

y=x2y = -x - 2

Would you like me to verify this equation or explain any part in more detail?


Related Questions:

  1. How do you calculate the slope if you are given two points?
  2. What is the general form of a linear equation, and how is it different from slope-intercept form?
  3. How can you identify the slope and yy-intercept directly from a graph?
  4. How do you convert the slope-intercept form into standard form?
  5. Can you explain how the slope relates to the steepness of the line?

Tip: Always double-check that the line passes through known points after calculating the equation.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Graphing Lines

Formulas

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

Theorems

Slope of a line between two points

Suitable Grade Level

Grades 8-10