Math Problem Statement

A line has a slope of 7. For each pair of points in problems 9 to 12 that are on the line, use the slope formula to find the missing coordinate. (x, 8) and (4, 1) : (3,-6) and (7,y)

Solution

To find the missing coordinates using the slope formula, we use the formula for the slope of a line:

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

Given that the slope m=7m = 7, we'll use this formula to solve for the missing coordinates.

Problem 1: (x,8)(x, 8) and (4,1)(4, 1)

The slope is 7, and we know the points (x,8)(x, 8) and (4,1)(4, 1). Using the slope formula:

7=184x7 = \frac{1 - 8}{4 - x}

This simplifies to:

7=74x7 = \frac{-7}{4 - x}

Now, solve for xx:

7(4x)=77(4 - x) = -7

287x=728 - 7x = -7

28+7=7x28 + 7 = 7x

35=7x35 = 7x

x=5x = 5

So, the missing coordinate for the first pair is x=5x = 5.

Problem 2: (3,6)(3, -6) and (7,y)(7, y)

Again, the slope is 7, and we know the points (3,6)(3, -6) and (7,y)(7, y). Using the slope formula:

7=y(6)737 = \frac{y - (-6)}{7 - 3}

This simplifies to:

7=y+647 = \frac{y + 6}{4}

Now, solve for yy:

7×4=y+67 \times 4 = y + 6

28=y+628 = y + 6

y=286y = 28 - 6

y=22y = 22

So, the missing coordinate for the second pair is y=22y = 22.

Summary:

  • For (x,8)(x, 8) and (4,1)(4, 1), x=5x = 5
  • For (3,6)(3, -6) and (7,y)(7, y), y=22y = 22

Would you like further details on the solution or any clarifications? Here are 5 related questions for practice:

  1. Find the slope between the points (2,3)(2, 3) and (5,15)(5, 15).
  2. Given the points (0,0)(0, 0) and (4,y)(4, y), and a slope of 2, find yy.
  3. Find the equation of the line passing through (2,5)(2, 5) and (4,11)(4, 11).
  4. If a line has a slope of 3 and passes through (1,4)(1, 4), find its equation in slope-intercept form.
  5. Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), what happens to the slope if x1=x2x_1 = x_2?

Tip: Remember, when two points have the same x-coordinates, the slope is undefined because you cannot divide by zero!

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Math Problem Analysis

Mathematical Concepts

Algebra
Coordinate Geometry
Linear Equations
Slope

Formulas

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

Theorems

Slope of a Line

Suitable Grade Level

Grades 8-10