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Chapter 8: Problem 14

Solve each system of equations. $$\begin{aligned} 4 x-2 y+z &=13 \\ 3 x-y+2 z &=13 \\ x+3 y-3 z &=-10 \end{aligned}$$

Short Answer

Expert verified
x = \frac{229}{17}, y = -\frac{6}{17}, z = -\frac{59}{17}

Step by step solution

01

Write the system of equations

Given the system of equations: \[ \begin{aligned} 4x - 2y + z &= 13 \ 3x - y + 2z &= 13 \ x + 3y - 3z &= -10 \end{aligned} \]
02

Eliminate one variable

To eliminate variable \( z \), manipulate the equations to align z-terms. Multiply the second equation by 2: \[ 2(3x - y + 2z) = 2(13) \] \ This gives: \[ 6x - 2y + 4z = 26 \] Subtract the first equation from this new equation: \[ (6x - 2y + 4z) - (4x - 2y + z) = 26 - 13 \] This simplifies to: \[ 2x + 3z = 13 \]
03

Substitute to simplify further

Subtract the third original equation from the first to eliminate z: \[ (4x - 2y + z) - (x + 3y - 3z) = 13 - (-10) \] Simplify: \[ 3x - 5y + 4z = 23 \] Now we have two new equations: \[ 2x + 3z = 13 \] \[ 3x - 5y + 4z = 23 \]
04

Solve the system with two unknowns

First solve for z in the equation: \[ 2x + 3z = 13 \] Express z: \[ z = \frac{13 - 2x}{3} \]. Substitute z into the second new equation: \[ 3x - 5y + 4\left(\frac{13 - 2x}{3}\right) = 23 \]. Simplify: \[ 3x - 5y + \frac{52 - 8x}{3} = 23 \], Multiply through by 3 to clear the fraction: \[ 9x - 15y + 52 - 8x = 69 \]. This simplifies to: \[ x - 15y + 52 = 69 \], Thus: \[ x - 15y = 17 \]
05

Solve for x and y

From the previous step: \[ x = 17 + 15y \]. Substitute x into one of the new equations: \[ 2(17 + 15y) + 3z = 13 \].Simplify: \[ 34 + 30y + 3z = 13 \]. Thus, \[ 3z = 13 - 34 - 30y \]. Solve: \[ z = -7 - 10y \].
06

Find values for all variables

We know: \[ x = 17 + 15y \] \[ z = -7 - 10y \]. Substitute these into the third original equation: \[ (17 + 15y) + 3y - 3(-7 - 10y) = -10 \]Simplify:\[ 17 + 15y + 3y + 21 + 30y = -10 \], Thus: \[ 68y + 38 = -10 \], Solve for y: \[ 68y = -48 \] \[ y = -\frac{24}{34} = -\frac{6}{17} \]
07

Determine x and z

Using the calculated y: \[ x = 17 + 15\left(-\frac{6}{17}\right) = 17 - \frac{90}{17} = \frac{289 - 90}{17} = \frac{229}{17} \]. Similarly for z: \[ z = -7 - 10\left(-\frac{6}{17}\right) = -7 + \frac{60}{17} = \frac{-119 + 60}{17} = -\frac{59}{17} \]

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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 mathematical expressions that describe relationships between variables using straight lines when graphed on a coordinate plane. Each equation in a system can be written in the general form: \[ ax + by + cz + ... = d \] Here, \(a, b, c\)... are constants (numbers), and \(x, y, z\)... are variables (unknown values to be determined). In the given system, we have three equations involving three variables \(x, y,\ and \ z\). The goal is to find the values of these variables that satisfy all the equations simultaneously.
Elimination Method
The elimination method involves combining equations to eliminate one of the variables, thus simplifying the system step by step. To apply this method effectively:
  • Align like terms (variables) in each equation
  • Multiply equations if necessary to make the coefficients of the chosen variable the same, or additive inverses
  • Add or subtract the equations to eliminate that variable
In our exercise, we chose to eliminate \(z\) by manipulating the second equation and then subtracting it from the first. This produced a new, simpler system of equations with fewer variables. Repeating this process systematically allows us to tackle even complex systems by breaking them down into easier parts.
Substitution Method
The substitution method involves solving one of the equations for one variable in terms of the others, and then substituting this expression into the remaining equations. Follow these steps to use substitution:
  • Solve one equation for one variable
  • Substitute this expression into the other equations
  • Simplify and solve the new equation
  • Repeat until all variables are determined
In our solution, after eliminating \(z\), we solved the resulting simpler equation for one variable (\(z\) in this case). This value was substituted back into the original equations, enabling us to find \(x\) and \(y\). By repeatedly substituting known values, we can determine each variable step-by-step.
Variables
Variables are symbols like \(x, y, z\) that represent unknown values. In systems of equations, these variables are what we need to solve for. The number of variables usually matches the number of equations in the system. Each variable in the equations represents a value that can change depending on the given constraints and relationships described by the equations.
  • Variables help form relationships and show how one quantity depends on others
  • Solving the system means finding the exact value for each variable
  • Different methods, such as elimination and substitution, can be used to isolate and determine these values
In our problem, identifying and manipulating variables through the system of equations was essential to find the unique set of values \(x, y,\ and \ z\) that satisfy each equation.

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Most popular questions from this chapter

Sketch the graph of the solution set to each nonlinear system of inequalities. $$\begin{aligned}&|x-1|<2\\\&|y-1|<4\end{aligned}$$

Solve each problem by using a system of three linear equations in three variables. Misplaced IIouse Numbers Angelo on Elm Street removed his three-digit house number for painting and noticed that the sum of the digits was 9 and that the units digit was three times as large as the hundreds digit. When the painters put the house number back up, they reversed the digits. The new house number was now 396 larger than the correct house number. What is Angelo's correct address?

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 2008 Gerhart lost twice as much in the futures market as he did in the stock market. If his losses totaled \(\$ 18,630,\) then how much did he lose in each market?

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. The Valley Health Club sold a dozen memberships in one week for a total of \(\$ 6000 .\) If male memberships cost \(\$ 500\) and female memberships cost \(\$ 500\), then how many male memberships and how many female memberships were sold?

Use a graphing calculator to graph the equation corresponding to each inequality. From the display of the graphing calculator, locate one ordered pair in the solution set to the system and check that it satisfies all inequalities of the system. Answers may vary. $$\begin{aligned}&y2^{x}\\\&y<5 x-6\end{aligned}$$
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