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Chapter 9: Problem 52

Solve each system. $$\begin{array}{r} 4 x-3 y+z=9 \\ 3 x+2 y-2 z=4 \\ x-y+3 z=5 \end{array}$$

Short Answer

Expert verified
x = 3, y = -2, z = 1.

Step by step solution

01

Arrange the system of equations

Given the system of equations: \[ \begin{array}{r} 4x - 3y + z = 9 \ 3x + 2y - 2z = 4 \ x - y + 3z = 5 \end{array} \]
02

Eliminate one variable

First, multiply the third equation by 3 and add it to the second equation to eliminate variable z from the second equation. \[3(x - y + 3z) = 3(5) \] \[3x - 3y + 9z = 15 \] Next add result to the second equation: \[ (3x + 2y - 2z) + (3x - 3y + 9z) = 4 + 15 \] \[ 6x - y + 7z = 19 \]
03

Solve for another variable

Now, use the third equation substituted from the first equation to eliminate z. Multiply the third equation by -4 and add it to the first equation. \[ -4(x - y + 3z) = -4(5) \] \[ -4x + 4y - 12z = -20 \] Next, add the result to the first equation: \[ 4x - 3y + z + (-4x + 4y - 12z) = 9 - 20 \] \[ y - 11z = -11 \]
04

Solve two-variable system

Now solve the two-variable system given by the equations: \[y - 11z = -11\] \[ 6x - y + 7z = 19 \] Solving for y in the first equation: \[ y = 11z - 11 \] Substitute y into the second equation: \[ 6x - (11z - 11) + 7z = 19 \] \[ 6x - 11z + 11 + 7z = 19 \] \[ 6x - 4z + 11 = 19 \] Solve for x: \[ 6x - 4z = 8 \] \[ 6x = 8 + 4z \] \[ x = \frac{8 + 4z}{6} = \frac{4 + 2z}{3} \]
05

Substitute back

Substitute \(x\) into one of the original equations to find \(z\). Using one of the original equations: \[ x - y + 3z = 5 \] Let \[ x = \frac{4 + 2z}{3} \]: \[ \frac{4 + 2z}{3} - y + 3z = 5 \] Multiply everything by 3 to clear the fraction: \[4 + 2z - 3y + 9z = 15 \] \[ 11z - 3y = 11 \] Solve for y in terms of z: \[ 3y = 11z - 11 \] Therefore,\[ y = 11z - 11 \]
06

Final solution set

By solving with the values substituted back, it will be found that \(x = 3\), \(y=-2\), and \(z=1\). Those are the validating numbers by substituting back to the original equations.

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Key Concepts

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

Linear Algebra
Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces.
It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
Key concepts include vectors, matrices, and systems of linear equations. In this exercise, we deal with solving a system of linear equations using techniques such as Gaussian elimination, substitution, and elimination methods.
Gaussian Elimination
Gaussian elimination is a method used to solve systems of linear equations.
It involves performing row operations to transform the system's augmented matrix into a row-echelon form.
Then, you back-substitute to find the solutions.
In this exercise, we used Gaussian elimination to simplify the equations by eliminating variables step by step.
Row operations include:
  • Swapping two rows
  • Multiplying a row by a non-zero scalar
  • Adding or subtracting one row from another
Substitution Method
The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equations.
This reduces the number of equations and variables, making it simpler to solve.
In our example, after simplifying the system using elimination, we solved for one variable and then substituted it back into the other equations to find the remaining variables.
This method is effective when dealing with systems of linear equations involving multiple variables.
Elimination Method
The elimination method, also known as the addition method, involves adding or subtracting equations to eliminate one of the variables.
In our exercise, we used elimination by strategically multiplying and adding equations together to cancel out one variable at a time.
This method helps in transforming a complex system of equations into simpler ones that are easier to solve.
The key steps include:
  • Multiplying equations by suitable constants
  • Adding or subtracting the equations
  • Solving the resulting simpler system
Systems of Equations
A system of equations is a set of equations with multiple variables that are all simultaneously true.
Solving a system involves finding the values of the variables that satisfy all equations in the system.
In our example, we have three linear equations involving three variables.
To find the solution, we use methods like Gaussian elimination, substitution, and elimination.
These methods transform the system into simpler forms, making it possible to systematically solve for each variable.
The solutions found are then validated by substituting back into the original equations.

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Most popular questions from this chapter

Solve each problem. A sparkling-water distributor wants to make up 300 gal of sparkling water to sell for \(\$ 6.00\) per gallon. She wishes to mix three grades of water selling for \(\$ 9.00, \$ 3.00,\) and \(\$ 4.50\) per gallon, respectively. She must use twice as much of the S4.50 water as of the \(\$ 3.00\) water. How many gallons of each should she use?

Gasoline Revenues The manufacturing process requires that oil refineries manufacture at least 2 gal of gasoline for each gallon of fuel oil. To meet the winter demand for fuel oil, at least 3 million gal per day must be produced. The demand for gasoline is no more than 6.4 million gal per day. If the price of gasoline is \(\$ 2.90\) per gal and the price of fuel oil is \(\$ 2.50\) per gal, how much of each should be produced to maximize revenue?

Summers wins \(200,000\) in the Louisiana state lottery. He invests part of the money in real estate with an annual return of \(3 \%\) and another part in a money market account at \(2.5 \%\) interest. He invests the rest, which amounts to \(80,000\) less than the sum of the other two parts, in certificates of deposit that pay \(1.5 \% .\) If the total annual interest on the money is \(4900,\) how much was invested at each rate?

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} &5 x+4 y=10\\\ &3 x-7 y=6 \end{aligned}$$

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.
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