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Chapter 1: Problem 18

Find an equation of the line passing through the given points. $$ (2,7) \text { and }(6,6) $$

Short Answer

Expert verified
Equation: \( y = -\frac{1}{4}x + 7.5 \)

Step by step solution

01

Find the Slope

First, use the formula for the slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] \[ m = \frac{6 - 7}{6 - 2} = \frac{-1}{4} = -\frac{1}{4} \]
02

Use Point-Slope Form

Next, use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Choose one of the points \(2, 7\): \[ y - 7 = -\frac{1}{4}(x - 2) \]
03

Simplify the Equation

Simplify the equation to put it in slope-intercept form \(y = mx + b\): \[ y - 7 = -\frac{1}{4}(x - 2) \] \[ y - 7 = -\frac{1}{4}x + \frac{1}{2} \] \[ y = -\frac{1}{4}x + \frac{1}{2} + 7 \] \[ y = -\frac{1}{4}x + 7.5 \]

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Key Concepts

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

finding slope
The slope of a line measures its steepness. To find the slope between two points, use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \((2, 7)\) and \((6, 6)\) are our points. Substituting these values: \[ m = \frac{6 - 7}{6 - 2} = \frac{-1}{4} \]. Thus, the slope, \( m \), is \(-\frac{1}{4} \).

This negative slope implies that the line is decreasing. Each unit step to the right corresponds to a \(1/4\)-unit step down.
point-slope form
The point-slope form is very useful when you know a point on the line and its slope. The formula is: \[ y - y_1 = m(x - x_1) \].

Let's use one of our points, \((2, 7)\), and the slope \( m = -\frac{1}{4} \). Substituting these values: \[ y - 7 = -\frac{1}{4}(x - 2) \].

This form allows you to find any point on the line given \( x \). It's a bridge to more familiar forms of a line's equation.
slope-intercept form
Slope-intercept form is one of the most recognizable ways to write an equation of a line: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

From the point-slope form \( y - 7 = -\frac{1}{4}(x - 2) \), we distribute and simplify: \[ y - 7 = -\frac{1}{4}x + \frac{1}{2} \] \[ y = -\frac{1}{4}x + \frac{1}{2} + 7 \] \[ y = -\frac{1}{4}x + 7.5 \].

Now, the equation shows the line's slope \( m = -\frac{1}{4} \) and its y-intercept \( b = 7.5 \).

This form can help you quickly graph the line and understand its behavior.

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Most popular questions from this chapter

In Exercises \(8-11\) , you are given a slope and a point on a line. Find another point on the same line. Then draw the line on graph paper. $$ \frac{1}{4} ; \text { point }(4,5) $$

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=3 x-3 $$

For each set of equations, tell what the graphs of all four relationships have in common without drawing the graphs. Explain your answers. $$\begin{array}{l}{y=-1.1 x+1.5} \\ {y=-1.1 x-4} \\ {y=-1.1 x+7} \\ {y=-1.1 x}\end{array}$$

Use the distributive property to rewrite each expression without using parentheses. \(p q\left(\frac{1}{p^{2}}-\frac{q}{p}\right)\)

The lines for these three equations all pass through a common point. $$y=\frac{x}{2}-1 \quad y=-\frac{2 x}{3}+6 \quad y=-\frac{x}{6}+3$$ a. Draw graphs for the three equations, and find the common point. b. Verify that the point you found satisfies all three equations by substituting the x- and y-coordinates into each equation.
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