Math Problem Statement

Write an equation in slope-intercept form of the line.

Solution

To write the equation of the line shown in the image in slope-intercept form, we need two pieces of information:

  1. The slope of the line.
  2. The y-intercept (the point where the line crosses the y-axis).

Step 1: Identify two points on the line

Looking at the graph, the line passes through two clear points:

  • Point 1: (10,5)(-10, 5)
  • Point 2: (10,1)(10, -1)

Step 2: Calculate the slope

The slope mm is calculated using the formula:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the coordinates of the two points: m=1510(10)=620=310m = \frac{-1 - 5}{10 - (-10)} = \frac{-6}{20} = -\frac{3}{10}

Step 3: Determine the y-intercept

To find the y-intercept bb, we use the slope-intercept form of the line equation:

y=mx+by = mx + b

Substitute the slope m=310m = -\frac{3}{10} and one of the points, say (10,1)(10, -1), into the equation: 1=310(10)+b-1 = -\frac{3}{10}(10) + b

Simplifying: 1=3+b    b=1+3=2-1 = -3 + b \implies b = -1 + 3 = 2

Step 4: Write the equation

Now that we have the slope m=310m = -\frac{3}{10} and the y-intercept b=2b = 2, the equation of the line is:

y=310x+2y = -\frac{3}{10}x + 2

This is the equation in slope-intercept form.

Would you like any further details or explanations?

Here are 5 related questions to deepen your understanding:

  1. How do you calculate the slope if you have two points?
  2. What does the y-intercept represent in a linear equation?
  3. How would the equation change if the slope was positive?
  4. Can you find the equation of a line if the y-intercept is not given but two points are?
  5. How would you write the equation if it were in point-slope form instead?

Tip: When calculating the slope, always subtract the coordinates in the same order (i.e., always y2y1y_2 - y_1 and x2x1x_2 - x_1) to avoid sign errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

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

Theorems

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Suitable Grade Level

Grades 8-10