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Chapter 8: Problem 59

Solve each problem by using a system of three linear equations in three variables. Just Numbers The sum of three numbers is \(40 .\) The difference between the largest and the smallest is \(12,\) and the largest is equal to the sum of the two smaller numbers. Find the numbers.

Short Answer

Expert verified
The numbers are 16, -4, and 28.

Step by step solution

01

Define the variables

Let the three numbers be represented by the variables: Let the smallest number be represented by \(x\),Let the middle number be represented by \(y\),Let the largest number be represented by \(z\).
02

Set up the equations based on the given information

The problem gives us three pieces of information that can be translated into equations:1. The sum of the three numbers is 40: \[ x + y + z = 40 \]2. The difference between the largest and the smallest number is 12: \[ z - x = 12 \]3. The largest number is equal to the sum of the two smaller numbers: \[ z = x + y \]
03

Solve the system of equations 鈥� substitute and simplify

Rearrange the second equation to solve for \(z\):\[ z = x + 12 \]Substitute this expression for \(z\) in the other two equations: \[ x + y + (x + 12) = 40 \]Simplify this equation: \[ 2x + y + 12 = 40 \]Subtract 12 from both sides:\[ 2x + y = 28 \]
04

Solve for one variable

Using the equation from Step 3, solve for \(y\):\[ y = 28 - 2x \]Substitute \(z\ = x + 12\) and \( y = 28 - 2x \) into the third equation (\(z = x + y\)):\[ x + 28 - 2x = x + 12 \]Simplify to find \(x\): \[ 28 - x = 12 \]\[ x = 16 \]
05

Find the other variables

Using \( x = 16 \) substitute back into the equations to find \(y\) and \(z\):\[ y = 28 - 2(16) \]\[ y = 28 - 32 \]\[ y = -4 \]Next, solve for \(z\):\[ z = x + 12 \]\[ z = 16 + 12 \]\[ z = 28 \]
06

Verify the values

Check the values of \(x\), \(y\), and \(z\) in all original equations to ensure they satisfy all conditions:\[ x + y + z = 40 \]\[ 16 - 4 + 28 = 40 \]\[ z - x = 12 \]\[ 28 - 16 = 12 \]\[ z = x + y \]\[ 28 = 16 - 4 \]All three equations are satisfied, so \(x = 16\), \(y = -4\), and \(z = 28\).

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Key Concepts

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

Solving Linear Equations
Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. Solving linear equations involves finding the values of the variables that make the equation true.
For example, in our given exercise, we have three linear equations:
1. The sum of three numbers is 40: \[ x + y + z = 40 \]
2. The difference between the largest and the smallest number is 12: \[ z - x = 12 \]
3. The largest number equals the sum of the two smaller numbers: \[ z = x + y \]
To solve these equations, we will manipulate them to isolate each variable, a process called solving.
Three Variables
When dealing with systems of equations, having three variables means we're working with equations that contain three unknowns, in this case, represented as \( x, y, \) and \( z \). The goal is to find values for these variables that satisfy all the given equations simultaneously.
In the context of our problem:
  • \( x \) represents the smallest number.
  • \( y \) represents the middle number.
  • \( z \) represents the largest number.
By forming equations based on the problem's statements and solving them step-by-step, we can determine the values of \( x, y, \) and \( z \).
Substitution Method
The substitution method is a technique used to solve systems of equations where you solve one equation for one variable and then substitute that expression into the other equation(s).
Here's how it works in our exercise:
  • First, we solve the second equation \( z = x + 12 \) to express \( z \) in terms of \( x \).
  • Next, we substitute this expression into the other equations to reduce the number of variables.
  • This results in simpler equations we can solve step-by-step.
This method simplifies the system for easier management and allows us to isolate each variable one at a time.
Linear Algebra
Linear algebra is a branch of mathematics focused on vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Understanding the principles of linear algebra is crucial in solving systems of equations like the one in our exercise.
In our problem, the three given linear equations form a system that can be represented in matrix form or solved using algebraic methods. While matrices and determinants are more advanced topics in linear algebra, for now, we focus on applying basic algebraic techniques:
  • Substitution
  • Combining like terms
  • Simplifying equations
This foundational understanding of linear algebra allows us to approach and solve more complex mathematical problems systematically.

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Most popular questions from this chapter

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. In Sociology 410 there are 55 more males than there are females. Two-thirds of the males and two-thirds of the females are graduating seniors. If there are 30 more graduating senior males than graduating senior females, then how many males and how many females are in the class?

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. One plan for federal income tax reform is to tax =an individual's income in excess of \(\$ 15,000\) at a \(17 \%\) rate. Another plan is to institute a national retail sales tax of \(15 \% .\) If =an individual spends \(75 \%\) of his or her income in retail stores where it is taxed at \(15 \%\), then for what income would the =amount of tax be the same under either plan?

Write a system of inequalities that describes the possible solutions to each problem and graph the solution set to the system. Size Restrictions United Parcel Service defines the girth of a box as the sum of the length, twice the width, and twice the height. The maximum girth that UPS will accept is 130 in. If the length of a box is 50 in., then what inequality must be satisfied by the width and height? Draw a graph showing the acceptable widths and heights for a length of 50 in.

Sketch the graph of the solution set to each nonlinear system of inequalities. $$\begin{aligned}&y<4-|x|\\\&y \geq|x|-4\end{aligned}$$

Solve each system of inequalities. $$\begin{aligned}&x^{2}+y^{2} \geq 9\\\&x^{2}+y^{2} \leq 25\\\&y \geq|x|\end{aligned}$$
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