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Chapter 3: Problem 6

Solve. The sum of three numbers is \(5 .\) The first number minus the second plus the third is \(1 .\) The first minus the third is 3 more than the second. Find the numbers.

Short Answer

Expert verified
The numbers are 4, 2, and -1.

Step by step solution

01

Define Variables

Let's define the three unknown numbers as variables: Let the first number be denoted as \( x \), the second number as \( y \), and the third number as \( z \).
02

Write Equations from Given Conditions

From the problem statement, we can write the following equations: 1. The sum of the three numbers is 5: \[ x + y + z = 5 \]2. The first number minus the second plus the third is 1: \[ x - y + z = 1 \]3. The first number minus the third is 3 more than the second: \[ x - z = y + 3 \]
03

Simplify Third Equation

Rewrite the third equation for easy substitution: \[ x - z = y + 3 \] becomes \[ x - y - z = 3 \]
04

Set Up System of Equations

Now our system of equations is: 1. \( x + y + z = 5 \)2. \( x - y + z = 1 \)3. \( x - y - z = 3 \)
05

Solve for One Variable

Subtract equation 2 from equation 1 to eliminate \( y \): \[ (x + y + z) - (x - y + z) = 5 - 1 \] \[ 2y = 4 \] \[ y = 2 \]
06

Substitute \( y \) into Equation 2

Substitute \( y = 2 \) into equation 2: \[ x - 2 + z = 1 \] \[ x + z = 3 \]
07

Substitute \( y \) into Equation 3

Substitute \( y = 2 \) into equation 3: \[ x - 2 - z = 3 \] \[ x - z = 5 \]
08

Solve for \( x \) and \( z \)

Now you have two new equations: 1. \( x + z = 3 \)2. \( x - z = 5 \) Add these two equations: \[ (x + z) + (x - z) = 3 + 5 \] \[ 2x = 8 \] \[ x = 4 \] Substitute \( x \) back into \( x + z = 3 \): \[ 4 + z = 3 \] \[ z = -1 \]
09

State the Solution

The three numbers are: \( x = 4 \), \( y = 2 \), and \( z = -1 \).

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Key Concepts

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

Linear Equations
Linear equations are a fundamental part of algebra and are used to express relationships between variables.
In this exercise, we are dealing with three linear equations:
  • The sum of three numbers: \( x + y + z = 5 \)
  • The first number minus the second plus the third: \( x - y + z = 1 \)
  • The first number minus the third is 3 more than the second: \( x - z = y + 3 \)
Each equation represents a straight line when graphed on a coordinate plane. These lines help in determining the points where variables intersect. The essence of solving linear equations is finding the values at which these intercepts happen.
Variable Substitution
Variable substitution is a method used to solve systems of equations by expressing one variable in terms of another.
It simplifies the system and makes it easier to find a solution.
In this exercise, we used substitution multiple times:
  • From \( y \): Substitute \( y = 2 \) back into the equations.
  • In Equation 2: \( x - 2 + z = 1 \)
  • In Equation 3: \( x - 2 - z = 3 \)
This transformed the system into simpler forms, making it easier to isolate variables and solve the equations.
Equation Simplification
Simplifying equations is crucial in solving complex problems.
It involves combining like terms and reducing the equation to its simplest form.
For instance, transforming \( x - z = y + 3 \) to \( x - y - z = 3 \) helped in straightforward substitution.
This step helps in removing any extra variables and focusing on solving one variable at a time.
Simplification ensures the equations are balanced and manageable, paving the way to finding correct solutions.
Problem-Solving Steps
Solving systems of equations requires a systematic approach.
Here are the key steps followed in this exercise:
  • Define Variables: \( x \), \( y \), and \( z \) represent the numbers.
  • Write Equations: Translate the problem's conditions into mathematical expressions.
  • Set Up System of Equations: Organize the equations for easier manipulation.
  • Solve for One Variable: Eliminate \( y \) by subtracting equations.
  • Substitute and Simplify: Replace solved values back into other equations to find remaining variables.
  • Solve for Remaining Variables: Use simplified equations to find \( x \) and \( z \).
  • State the Solution: Clearly outline the values found for each variable. The final answers are: \( x = 4 \), \( y = 2 \), and \( z = -1\)
Following these steps helps in breaking down the problem, making it manageable and ensuring an accurate solution.

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Most popular questions from this chapter

Brushstroke Computers, Inc., is planning a new line of computers, each of which will sell for \(\$ 970 .\) The fixed costs in setting up production are \(\$ 1,235,580,\) and the variable costs for each computer are \(\$ 697 .\) a) What is the break-even point? b) The marketing department at Brushstroke is not sure that \(\$ 970\) is the best price. Their demand function for the new computers is given by \(D(p)=-304.5 p+374,580\) and their supply function is given by \(S(p)=788.7 p-\) \(576,504 .\) To the nearest dollar, what price \(p\) would result in equilibrium between supply and demand? c) If the computers are sold for the equilibrium price found in part \((b),\) what is the break-even point?

Automobile Pricing. The base model of a 2016 Jeep Wrangler Sport with a tow package cost 24,290 dollars. When equipped with a tow package and a hard top, the vehicle's price rose to 25,285 dollars. The cost of the base model with a hard top was 24,890 dollars. Find the base price, the cost of a tow package, and the cost of a hard top. Data: jeep.com

Dana is designing three large perennial flower beds for her yard and is planning to use combinations of three types of flowers. In her traditional cottage-style garden, Dana will include 7 purple coneflower plants, 6 yellow foxglove plants, and 8 white lupine plants at a total cost of 93.31 dollars. The flower bed around her deck will be planted with 12 yellow foxglove plants and 12 white lupine plants at a total cost of 126.00 dollars. A third garden area in a corner of Dana's yard will contain 4 purple coneflower plants, 5 yellow foxglove plants, and 7 white lupine plants at a total cost of 72.82 dollars. What is the price per plant for the coneflowers, the foxgloves, and the lupines?

Bakery. Gigi's Cupcakes offers a gift box with six cupcakes for $$ 15.99 .\( Gigi's also sells cupcakes individually for \)3 each. Gigi's sold a total of 256 cakes one Saturday for a total of $$ 701.67$ in sales (excluding tax). How many six-cupcake gift boxes were included in that day's sales total?

Many graduate schools require applicants to take the Graduate Record Examination (GRE). Those taking the GRE receive three scores: a verbal reasoning score, a quantitative reasoning score, and an analytical writing score. In 2013 , the average quantitative reasoning score exceeded the average verbal reasoning score by 1.6 points, and the average verbal reasoning score exceeded the analytical writing score by 147.1 points. The sum of the three average scores was \(306.3 .\) What was the average score for each category? Data: Educational Testing Service
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