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Chapter 3: Problem 24

A line passes through the given points. (a) Find the slope of the line. (b) Write the equation of the line in slope-intercept form. $$ (10,15) ;(6,27) $$

Short Answer

Expert verified
The equation of the line is \( y = -3x + 45 \)

Step by step solution

01

- Identify the coordinates

Given points are \(x_1, y_1\) = (10, 15) and \(x_2, y_2\) = (6, 27).
02

- Calculate the slope

Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the coordinates into the formula: \[ m = \frac{27 - 15}{6 - 10} = \frac{12}{-4} = -3 \]
03

- Use the slope-intercept form

The slope-intercept form of a line is given by: \[ y = mx + b \] Here, \(m = -3\). Substitute one point to solve for \(b\). Using point \( (10, 15) \) we get: \[ 15 = -3(10) + b \] Simplify to solve for \( b \): \[ 15 = -30 + b \] \[ b = 45 \]
04

- Write the equation

Substitute \(m = -3\) and \(b = 45\) into the slope-intercept form: \[ y = -3x + 45 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Formula
The slope of a line measures its steepness and direction. The slope is represented by the letter 'm'. To find the slope between two points, \(x_1, y_1\) and \(x_2, y_2\), we use the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this exercise, the points given are (10, 15) and (6, 27). By substituting these coordinates into the formula, we get:
\[ m = \frac{27 - 15}{6 - 10} = \frac{12}{-4} = -3 \] This calculation shows that the slope of the line passing through these points is -3.

Understanding the slope is essential as it tells us about the line's inclination:
  • A positive slope indicates the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • A slope of zero means the line is horizontal.
  • An undefined slope means the line is vertical.
Slope-Intercept Form
The slope-intercept form of a linear equation is \[ y = mx + b \] where:
  • \

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Most popular questions from this chapter

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{4} x+3\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(-8, \frac{1}{4}\right) ;\left(16, \frac{1}{2}\right)\)

Use the slope formula to find the slope of the line that passes through the points. \((-3,8) ;(11,23)\)

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=2 x+5\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-4 y=48\)
See all solutions

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