Math Problem Statement

Write the equation of the line that passes through the points left parenthesis, 9, comma, minus, 1, right parenthesis(9,−1) and left parenthesis, minus, 7, comma, 4, right parenthesis(−7,4). Put your answer in fully simplified point-slope form, unless it is a vertical or horizontal line.

Solution

To find the equation of the line passing through the points (9,1)(9, -1) and (7,4)(-7, 4), we follow these steps:

1. Find the slope of the line (denoted by mm):

The formula for the slope mm between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

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

Substituting the given points (9,1)(9, -1) and (7,4)(-7, 4):

m=4(1)79=4+179=516=516m = \frac{4 - (-1)}{-7 - 9} = \frac{4 + 1}{-7 - 9} = \frac{5}{-16} = -\frac{5}{16}

2. Use the point-slope form of the equation of a line:

The point-slope form is given by:

yy1=m(xx1)y - y_1 = m(x - x_1)

Using the point (9,1)(9, -1) and the slope m=516m = -\frac{5}{16}:

y(1)=516(x9)y - (-1) = -\frac{5}{16}(x - 9)

Simplifying the equation:

y+1=516(x9)y + 1 = -\frac{5}{16}(x - 9)

This is the equation of the line in point-slope form.

Would you like more details on solving it further, or have any questions?

Here are 5 related questions:

  1. How do you convert from point-slope form to slope-intercept form?
  2. What happens when the line is vertical or horizontal?
  3. How can you check if a point lies on a line?
  4. How do you find the distance between two points?
  5. How do you find the equation of a line parallel to a given line?

Tip: The slope of a vertical line is undefined, while a horizontal line has a slope of zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Point-Slope Form
Slope of a Line

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Point-slope form: y - y1 = m(x - x1)

Theorems

Slope-Intercept Theorem

Suitable Grade Level

Grades 8-10