/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 In Exercises \(23-24\), let \(x\... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 24

In Exercises \(23-24\), let \(x\) represent the first number, \(y\) the second number, and \(z\) the third number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The following is known about three numbers: Three times the first number plus the second number plus twice the third number is \(5 .\) If 3 times the second number is subtracted from the sum of the first number and 3 times the third number, the result is \(2 .\) If the third number is subtracted from 2 times the first number and 3 times the second number, the result is \(1 .\) Find the numbers.

Short Answer

Expert verified
The values of the numbers \(x, y, z\) are 1, 2 and 3 respectively.

Step by step solution

01

Translating Words into Equations

Based on the given conditions, we can create the following system of equations: \n\n1. \(3x + y + 2z = 5\)\n2. \(x + 3z - 3y = 2\)\n3. \(2x + 3y - z = 1\)
02

Solving the Equation

Use substitution or elimination to find values for \(x, y,\) and \(z\). Let's employ substitution: \n\nFrom equation (2), express \(x\) in terms of \(y\) and \(z\): \(x = 3y + 2 - 3z\). \n\nSubstitute \(x\) into equation (3):\n \(2(3y + 2 - 3z) + 3y - z = 1 \)\n Simplify: \(6y + 4 - 6z + 3y - z = 1 \)\n Put like terms together and simplify: \(9y - 7z = -3 \)\n\n Amend equation (1) to look like equation (4) by multiplying it by 3: \(9x + 3y + 6z = 15\)\n\nSubstitute equation (4) into improved equation (1):\n \(9(3y + 2 - 3z) + 3y + 6z = 15\)\n Simplify: \(27y + 18 - 27z + 3y + 6z = 15\)\n Put like terms together and simplify: \(30y - 21z = -3\)\n\nNow we have two new equations: \(9y - 7z = -3\) and \(30y - 21z = -3\). Solve them to find \(y\), and \(z\). Afterwards, we will find the value of \(x\).
03

Finding the Values

Multiply the first equation by 30 and the second by 9 \n\n270y - 210z = -90 (5)\n270y - 189z = -27 (6)\n\nSubtract equation (6) from (5) to get z:\n \n-21z = -63\nSo, z = 3\n\nSubstitute z = 3 into the equation 9y - 7z = -3 to get y:\n\n9y - 7(3) = -3\n9y - 21 = -3\nSo, y = 2\n\nSubstitute y = 2 and z = 3 into the equation x = 3y + 2 - 3z to get x:\n\nx = 3(2) + 2 - 3(3) \nSo, x = 1

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique used to solve systems of equations. It involves expressing one variable in terms of others, then substituting this expression into other equations in the system. This method helps you find the values of the unknowns step-by-step.
To use substitution, begin by isolating one of the variables in one of the equations. For example, from equation (2) in the original problem, we expressed \(x\) in terms of \(y\) and \(z\): \(x = 3y + 2 - 3z\).
Next, substitute this expression into another equation. We substituted \(x = 3y + 2 - 3z\) into equation (3), which led to:
  • \(2(3y + 2 - 3z) + 3y - z = 1\)
  • This further simplifies to \(9y - 7z = -3\)

By this substitution, we created a new equation with only two variables. This new simplified system can then be solved using the elimination method or repetition of substitution.
Elimination Method
The elimination method is another effective approach for solving systems of equations. It involves adding or subtracting equations to eliminate a variable, making it possible to solve for the remaining variables.
In the context of the original exercise, after using substitution, two new equations were formed:
  • \(9y - 7z = -3\)
  • \(30y - 21z = -3\)

To eliminate variables using the elimination method, align the equations so that adding them cancels out one variable. This may require multiplying one or both equations by a coefficient to align them.
In this exercise, equations were multiplied to make the coefficients of \(y\) equal, resulting in:
  • \(270y - 210z = -90\) (from multiplying the first equation by 30)
  • \(270y - 189z = -27\) (from multiplying the second equation by 9)

Subtracting these equations eliminates \(y\) and solves for \(z\). Afterwards, substitute back to find the other unknowns. This step-by-step elimination leads to finding the solution for all variables efficiently.
Solving Linear Equations
Solving linear equations is an essential algebraic skill which involves finding the values of unknown variables that make the equation true. It is at the heart of solving systems of equations.
Linear equations can typically be written in the form \(ax + by + cz = d\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\), \(y\), and \(z\) are variables. In the given exercise, each equation relates these concepts:
  • Equation 1: \(3x + y + 2z = 5\)
  • Equation 2: \(x + 3z - 3y = 2\)
  • Equation 3: \(2x + 3y - z = 1\)

The goal is to find a common solution for these equations. By converting the given information into solvable linear equations and applying methods like substitution or elimination, variable values can be systematically deduced.
Whether using substitution or elimination, the key to solving linear equations is breaking them down into simpler components. This simplification can then be used strategically to isolate and solve for each variable individually, building to the complete solution step by step.

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Most popular questions from this chapter

Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\)-pound weight restriction.

If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)

Determine whether cach statement makes sense or does not make sense, and explain your reasoning. Even if a linear system has a solution set involving fractions, such as \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\}, 1\) can use graphs to determine if the solution set is reasonable.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a nonlinear system that modeled the orbits of Earth and Mars, and the graphs indicated the system had a solution with a real ordered pair.

In \(1978,\) a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{lcc}\hline & \text { Boeing 727 } & \text { Falcon 20 } \\\\\hline \text { Direct Operating cost } & \$ 1400 \text { per hour } & \$ 500 \text { per hour } \\\\\text { Payload } & 42,000 \text { pounds } & 6000 \text { pounds }\end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to \(\$ 35,000 .\) \(\cdot\) Total payload had to be at least \(672,000\) pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?
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