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Chapter 4: Problem 1

The slope of the line between \((2,10)\) and \(\left(x_{2}, 4\right)\) is \(-3\). Find the value of \(x_{2}\).

Short Answer

Expert verified
The value of \( x_2 \) is 4.

Step by step solution

01

Understand the Concept of Slope

The slope of a line is a measure of its steepness and is calculated as the change in the y-values divided by the change in the x-values between two points on the line. Mathematically, this is represented as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We are given that the slope \( m \) is -3.
02

Write down the Slope Formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this problem, the points are \((2, 10)\) and \( (x_2, 4) \), and the slope \( m \) is -3. Let us write this out with the known values.
03

Substitute Known Values into the Formula

Substitute \((x_1, y_1) = (2, 10)\), \( (x_2, y_2) = (x_2, 4) \), and \( m = -3 \) into the slope formula: \[ -3 = \frac{4 - 10}{x_2 - 2} \].
04

Simplify the Equation

Calculate \( 4 - 10 = -6 \) and substitute into the equation: \[ -3 = \frac{-6}{x_2 - 2} \].
05

Solve for \( x_2 \)

To find \( x_2 \), multiply both sides of the equation by \( x_2 - 2 \) to eliminate the fraction: \[ -3(x_2 - 2) = -6 \].
06

Distribute and Simplify

Distribute -3 to \( x_2 - 2 \): \[ -3x_2 + 6 = -6 \].
07

Solve for \( x_2 \)

Subtract 6 from both sides: \[ -3x_2 = -12 \]. Then, divide by -3: \[ x_2 = 4 \].

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Key Concepts

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

Slope
The slope of a line is a key algebraic concept that defines the line's steepness. You can think of slope as a way to measure how "slanted" a line is. In mathematical terms, slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.
The formula for finding slope, denoted as \(m\), is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This formula helps you calculate the change in y-values divided by the change in x-values.
Consider the points (2, 10) and \((x_2, 4)\) given in the problem, where the slope is \(-3\). This means that for every unit you move horizontally, the line moves vertically by -3 units. This negative slope indicates the line is tilting downwards.
Linear Equations
Linear equations represent straight lines when graphed, forming a direct proportional relationship between the variables. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In this exercise, we manipulate a linear equation derived from the slope formula to solve for an unknown. We start with:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
After substituting known values (e.g., slope \(-3\), points (2, 10) and \((x_2, 4)\)), we arrive at the equation \(-3 = \frac{4 - 10}{x_2 - 2}\).
This exercise demonstrates the utility of linear equations in solving for an unknown variable and understanding relationships between variables.
Problem Solving
Problem solving in algebra often involves breaking down complex problems into simpler parts. In this exercise, we aim to find a missing coordinate using the concept of slope.
First, identify what you know:
  • The slope \(m\) is \(-3\).
  • The known points are (2, 10) and \((x_2, 4)\).
Substitute these into the slope formula: \(-3 = \frac{4 - 10}{x_2 - 2}\).
Next, simplify by calculating the numerator: \(4 - 10 = -6\). Then, solve the equation \(-3 = \frac{-6}{x_2 - 2}\). By multiplying through by \(x_2 - 2\) and solving for \(x_2\), we find that \(x_2 = 4\).
Effective problem solving requires understanding the relationship between variables, substituting values into formulas, and simplifying equations to find unknowns.

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Most popular questions from this chapter

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A 4 oz bottle of mustard costs \(\$ 0.88\), a \(7.5\) oz bottle costs \(\$ 1.65\), and an 18 oz bottle costs \(\$ 3.99\). Is the size of the mustard bottle directly proportional to the price? If so, show how you know. If not, suggest the change of one or two prices so that they will be directly proportional. (Ti)

At 2:00 P.M, elevator A passes the second floor of the Empire State Building going up. The table shows the floors and the times in seconds after 2:00. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Floor \(x\) & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\ \hline Time after 2:00 (s) \(y\) & 0 & \(1.3\) & \(2.5\) & \(3.8\) & 5 & \(6.3\) & \(7.5\) \\\ \hline \end{tabular} a. What is the line of fit based on Q-points for the data? b. Give a real-world meaning for the slope. (a) c. About what time will this elevator pass the 60th floor if it makes no stops? (a) d. Where will this elevator be at \(2: 00: 45\) if it makes no stops?

In Chapter 3 you worked with problems involving rate, often involving the equation \(d=r t\). Here is another kind of rate problem. Ellen and Eric meet on Saturday to train for a marathon. They live 7 miles apart and meet at the high-school track that is between their two homes. Ellen leaves at 8:00 A.M. and jogs south toward the school at \(4 \mathrm{mi} / \mathrm{h}\). Eric waits until 8:30 A.M. and jogs north toward the school at \(6 \mathrm{mi} / \mathrm{h}\). The two friends arrive at the school at exactly the same time. How much time did each person jog? To solve this problem, let \(t\) represent Ellen's time in hours. Because Eric left a half hour after Ellen, but arrived at the same time, he jogged for a half hour less. So let \(t-\frac{1}{2}\) represent Eric's time in hours. You might now fill out a table like this to get expressions for distance. (Remember that distance \(=\) rate \(\cdot\) time.) a. Write an equation that states that Ellen's distance and Eric's distance combine to 7 miles. b. Solve the equation from \(6 a\), check your answer, and state the solution. c. Solve this problem using a similar procedure: A propeller airplane and a jet airplane leave the same airport at the same time, and both go in the same direction. The jet airplane's velocity is five times the propeller airplane's velocity. After \(2.25 \mathrm{~h}\), the jet airplane is \(1170 \mathrm{~km}\) ahead of the propeller plane. What is the velocity of each plane in kilometers per hour?
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