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Chapter 4: Problem 73

Below are two tables of values for two linear equations. Using the tables, a. find a solution of the corresponding system. b. graph several ordered pairs from each table and sketch the two lines. c. Does your graph confirm the solution from part a? $$ \begin{array}{|c|c|} \hline x & {y} \\ \hline 1 & {3} \\ \hline 2 & {5} \\ \hline 3 & {7} \\ \hline 4 & {9} \\ \hline 5 & {11} \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline x & {y} \\ \hline 1 & {6} \\ \hline 2 & {7} \\ \hline 3 & {8} \\ \hline 4 & {9} \\ \hline 5 & {10} \\ \hline \end{array} $$

Short Answer

Expert verified
The solution to the system is (4, 9); the graphs intersect at this point, confirming the solution.

Step by step solution

01

Finding the Linear Equations

To solve the system, first find both linear equations. Look at the changes in the table values to determine the slope (m) for each equation. For the first table, as \( x \) increases by 1, \( y \) increases by 2, so the slope is 2. Using the point (1, 3), the equation is \( y = 2x + 1 \). For the second table, \( y \) increases by 1 as \( x \) increases by 1, so the slope is 1. Using the point (1, 6), the equation is \( y = x + 5 \).
02

Finding the Solution of the System

To find the solution, set the equations equal to each other: \( 2x + 1 = x + 5 \). Solve for \( x \) by subtracting \( x \) and 1 from both sides to get \( x = 4 \). Substitute \( x = 4 \) into one of the original equations, like \( y = 2x + 1 \), to get \( y = 9 \). So, the solution is \((4, 9)\).
03

Graphing the Equations

Graph several ordered pairs from each table on coordinate axes. For the first line \( y = 2x + 1 \), plot (1, 3), (2, 5), and (3, 7). For the second line \( y = x + 5 \), plot (1, 6), (2, 7), and (3, 8). Draw lines through each set of points. The intersection point should be at \((4, 9)\).
04

Confirming the Solution with the Graph

The graph confirms the solution from part a because both lines intersect at the point \((4, 9)\), which matches the calculated solution. This graphical verification ensures that the solution is correct and satisfies both equations simultaneously.

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Key Concepts

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

System of Equations
A system of equations consists of two or more equations that share common variables. In this exercise, we have two linear equations, each represented by a table of values. Each row in the tables provides an ordered pair \(x, y\) that satisfies the respective equation. The goal is to find a set of values for these variables that satisfy both equations simultaneously.
To solve a system of equations, we look for the point where both equations intersect each other. This point is called their "solution" and is represented as an ordered pair \(x, y\).
Simplifying, a system's solution is where both equations stay true: \(y_1 = y_2\), ensuring both equations use the same \(x\) and \(y\).
  • Understanding the relationship between variables in a system is crucial.
  • The solution is where all equations in the system are satisfied at once.
Graphical Solutions
Graphing equations is a helpful way to visually solve systems. Graphical solutions illustrate the relationship between the variables in each equation. As seen in this exercise, representing the equations as lines on a graph offers a clear visual understanding of where they intersect.
When plotting the equations on a graph:
  • The x-values and y-values from each table define points you plot on the graph's plane.
  • Once plotted, these points form lines based on the equations derived from the tables.
The point where the lines intersect is the graphical solution to the system. This intersection represents the set of coordinates that solve the system of equations. In our specific case, both lines intersect at the point \(4, 9\). As seen in the graph, this is the only location that satisfies both equations at once.
Solving Linear Systems
Solving linear systems involves finding a common solution to two or more linear equations. There are several methods to tackle these systems, such as substitution, elimination, and graphing.
In the provided exercise, we focused on graphical and algebraic methods to identify the solution. We started by finding the linear equations derived from the tables:
  • Equation 1: \( y = 2x + 1 \)
  • Equation 2: \( y = x + 5 \)
We then set these equations equal to each other to find the solution. Solving \( 2x + 1 = x + 5 \), we simplified to find \( x = 4 \). Substituting \( x = 4 \) back into one of the original equations allowed us to find \( y = 9 \).
This solution, \(4, 9\), represents the point where both equations intersect, confirming that it solves both equations simultaneously.
Understanding how these equations relate helps solidify the understanding of solving linear systems, ensuring clear paths to the correct solution.

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Most popular questions from this chapter

Solve. See a Concept Check in this section. The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014 , the number of men receiving degrees (in thousands) is given by the equation \(y=3.9 x+443\) and for women, the equation is \(y=14.2 x+314\) where \(x\) is the number of years after \(1970 .\) (Source: National Center for Education Statistics) (IMAGE CANNOT COPY) A. Use the substitution method to solve this system of equations. (Round your final results to the nearest whole numbers.) B. Explain the meaning of your answer to part (a). C. Sketch a graph of the system of equations. Write a sentence describing the trends for men and women receiving bachelor degrees.

The sum of the digits of a three-digit number is \(15 .\) The tens-place digit is twice the hundreds-place digit, and the ones-place digit is 1 less than the hundreds-place digit. Find the three-digit number.

Solve each system of linear equations by graphing. See Examples 3 through 6 $$ \left\\{\begin{array}{l} {3 x-y=6} \\ {\frac{1}{3} y=-2+x} \end{array}\right. $$

Given the cost function \(C(x)\) and the revenue function \(R(x)\), find the number of units \(x\) that must be sold to break even. See Example 6. $$ C(x)=30 x+10,000 R(x)=46 x $$

Davie and Judi Mihaly own 50 shares of Apple stock and 60 shares of Microsoft stock. At the close of the markets on March \(9,2007,\) their stock portfolio was worth 6035.90 dollars. The closing price of the Microsoft stock was 60.68 dollars less than the closing price of Apple stock on that day. What was the price of each stock on March \(9,2007 ?\) (Source: New York Stock Exchange)
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