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Chapter 2: Problem 301

Given the following set of information, find a linear equation satisfying the conditions, if possible. Passes through (5, 1) and (3, 鈥�9)

Short Answer

Expert verified
The linear equation is \( y = 5x - 24 \).

Step by step solution

01

Identify Two Points

The problem provides two points through which the line passes: (5, 1) and (3, -9). These points will allow us to calculate the slope and eventually the equation of the line.
02

Calculate the Slope

Use the formula for the slope, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points (5, 1) as \((x_1, y_1)\) and (3, -9) as \((x_2, y_2)\), we get: \[ m = \frac{-9 - 1}{3 - 5} = \frac{-10}{-2} = 5. \] Thus, the slope, \( m \), is 5.
03

Write the Equation in Point-Slope Form

We now use the point-slope form of the equation, \( y - y_1 = m(x - x_1) \), with one of the given points. Using the point (5, 1) and the slope \( m = 5 \), the equation becomes: \[ y - 1 = 5(x - 5). \]
04

Simplify to Slope-Intercept Form

Distribute and simplify the equation from the point-slope form to slope-intercept form, \( y = mx + b \):\[ y - 1 = 5x - 25 \] Add 1 to both sides to solve for \( y \):\[ y = 5x - 25 + 1 \]soy = 5x - 24. Thus, the slope-intercept form of the equation is \( y = 5x - 24 \).
05

Verify the Equation with Second Point

Substitute the second point (3, -9) into the equation \( y = 5x - 24 \) to verify: \[ -9 = 5(3) - 24 \]\[ -9 = 15 - 24 \]\[ -9 = -9 \] Since this statement is true, the equation is correctly fitted to both points.

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Key Concepts

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

Slope Calculation
Calculating the slope of a line is a crucial step in formulating a linear equation. The slope tells us how steep a line is and the direction it moves鈥攗p or down as we move along the x-axis. To find the slope (\( m \)), you need two points on the line, like the ones given in the problem. The formula for slope is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Using the points (5, 1) and (3, -9), assign (5, 1) as \( (x_1, y_1) \) and (3, -9) as \( (x_2, y_2) \). When you substitute into the formula, you get:
  • \( m = \frac{-9 - 1}{3 - 5} = \frac{-10}{-2} = 5 \)

Therefore, the slope is 5, indicating that for every one unit increase in x, y increases by 5 units. It is a positive slope, meaning the line rises as it goes from left to right on a graph.
Point-Slope Form
The point-slope form is particularly useful when you know a point on the line and the slope. This form is represented as:
  • \( y - y_1 = m(x - x_1) \)
Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. From the equation derived using the point (5, 1) and slope 5, it is written as:
  • \( y - 1 = 5(x - 5) \)

This form is useful because it directly incorporates both the slope and a specific point from the line. It provides a straightforward way to see if another point lies on the line by simply plugging it in.
Slope-Intercept Form
When we talk about the slope-intercept form, it's all about making equations easier to understand. Its general format is:
  • \( y = mx + b \)
Here, \( m \) is the slope, and \( b \) represents the y-intercept, the point where the line crosses the y-axis. We started from the point-slope form:
  • \( y - 1 = 5(x - 5) \)

By simplifying it, we convert it to slope-intercept form:
  • First, distribute the 5: \( y - 1 = 5x - 25 \)
  • Then, isolate y by adding 1: \( y = 5x - 24 \)
This gives you the final equation in slope-intercept form, \( y = 5x - 24 \). Now, both the slope (5) and y-intercept (-24) are clear.
Verifying Solutions
Verifying a solution involves checking if the equation derived holds true for all the points given. In our example, after forming the equation \( y = 5x - 24 \), we use the second point (3, -9) to ensure it fits the line.
  • Substitute \( x = 3 \): \(-9 = 5(3) - 24 \)
  • Simplify: \(-9 = 15 - 24 \)
  • Results in \(-9 = -9 \)

Since the original and calculated values equal each other, the line is confirmed to pass through the point (3, -9). This step reassures you that your equation and calculations are accurate, confirming the line's integrity to represent the given data.

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Most popular questions from this chapter

For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? Line \(1 :\) Passes through \((8,-10)\) and \((0,-26)\) Line \(2 :\) Passes through \((2,5)\) and \((4,4)\)

Find the value of \(x\) if a linear function goes through the following points and has the following slope: \((x, 2),(-4,6), m=3\)

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {900} & {988} & {1000} & {1010} & {1200} & {1205} \\ \hline y & {70} & {80} & {82} & {84} & {105} & {108} \\\ \hline\end{array}$$

Graph \(f(x)=-2 x-10 .\) Pick a set of 5 ordered pairs using inputs \(x=-2,1,5,6,9\) and use linear regression to verify the function.

For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in Table 2.14. Assume that the house values are changing linearly. $$\begin{array}{|c|c|c|}\hline \text { Year } & {\text { Pima Central }} & {\text { East valley }} \\ \hline 1970 & {32,000} & {120,250} \\ \hline 2010 & {85,000} & {150,000} \\ \hline\end{array}$$ If these trends were to continue, what would be the median home value in Pima Central in 2015?
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