Math Problem Statement

What is the slope of the line that passes through the points left parenthesis, 4, comma, minus, 6, right parenthesis(4,−6) and left parenthesis, minus, 2, comma, 3, right parenthesis(−2,3)? Write your answer in simplest form.

Solution

To find the slope of the line passing through two points, we use the slope formula:

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

Given the points (4,6)(4, -6) and (2,3)(-2, 3), we can assign:

  • x1=4x_1 = 4, y1=6y_1 = -6
  • x2=2x_2 = -2, y2=3y_2 = 3

Now, substitute these values into the formula:

m=3(6)24=3+624=96=32m = \frac{{3 - (-6)}}{{-2 - 4}} = \frac{{3 + 6}}{{-2 - 4}} = \frac{9}{-6} = -\frac{3}{2}

So, the slope of the line is:

m=32m = -\frac{3}{2}

Would you like more details or have any questions?

Here are five related questions for practice:

  1. What is the slope of a line passing through the points (2,5)(2, 5) and (3,4)(-3, -4)?
  2. How do you interpret the slope of 32-\frac{3}{2} in the context of this line?
  3. If the slope is positive, what does that tell you about the direction of the line?
  4. How do you find the y-intercept of a line given its slope and a point?
  5. Can two points define more than one unique line? Why or why not?

Tip: The slope of a line indicates its steepness and direction—positive slopes go upward, negative slopes go downward.

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Math Problem Analysis

Mathematical Concepts

Algebra
Coordinate Geometry
Slopes of Lines

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Linear Equation Properties
Slope-Intercept Form

Suitable Grade Level

Grades 8-10