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Chapter 5: Problem 21

Solve the system of equations. $$\left\\{\begin{array}{rr}x+4 z & =13 \\ 4 x-2 y+z= & 7 \\ 2 x-2 y-7 z & =-19\end{array}\right.$$

Short Answer

Expert verified
The solution of the system of equations is x = 13, y = 22.5, and z = 0.

Step by step solution

01

Simplify the equations

The first step is to make the equations easier to manage. Let's isolate 'x' in the first equation.\[x = 13 - 4z\]
02

Substitute x in the other equations

Substitute the obtained value of 'x' from step 1 into the remaining two equations:\[4(13 - 4z) - 2y + z = 7\]\[2(13 - 4z) - 2y - 7z = -19\]Now simplify these equations and isolate y. After simplifying, we get:\[-6z - 2y = -45\]\[-15z - 2y = -45\]This gives us a system of two new equations with two variables (y and z).
03

Solve the system of two equations

To find the values of 'y' and 'z', we can subtract the second equation from the first:\[9z = 0\]Solving this, we find z = 0. Substituting 'z' back into one of the two-variable equations, we can find 'y'. Let's use the first equation:\[-6(0) - 2y = -45\]Solving this equation, we find y = 22.5.
04

Find the value of 'x'

Substitute the values of 'y' and 'z' into the original first equation to find 'x':\[x = 13 - 4(0)\]So, x = 13.

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Key Concepts

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

Understanding Algebraic Solutions
An algebraic solution involves finding the values of variables that satisfy given equations. In this context, we are solving a system of linear equations, which consists of multiple equations that are related through common variables. The goal is to find a set of values for the variables that make all the equations true at the same time.

To achieve this, we manipulate the equations using policies such as substitution or elimination, eventually finding a single solution or determining that no solution exists. Each equation in the system forms a straight line when graphed on a coordinate plane, and the solution is the point where these lines intersect, representing the common values of the variables.
  • The solution involves simplifying equations by eliminating variables systematically.
  • We strive to get each equation in terms of a single variable to solve them one by one, which facilitates tracking and solving simultaneously multiple variables.
Exploring the Substitution Method
The substitution method is one of the core approaches to solving systems of equations. It involves expressing one variable in terms of the others and substituting this expression into the other equations in the system.

In our specific example, we started by solving for 'x' in the first equation. By isolating 'x', we get: \[ x = 13 - 4z \]. This expression for 'x' is then placed into the other equations, simplifying the system to focus on the remaining variables, in this case, 'y' and 'z'. This method is particularly useful when one of the equations is easily solvable for one of the variables, as it simplifies the process of dealing with more complex systems.
  • Prioritize isolating a variable that simplifies the procedure.
  • Efficiency increases with simpler substitution equations.
The Basics of Linear Equations
Linear equations form the basis of the system we are solving. A linear equation is any equation that graphs to a straight line and is formed by constants and variables raised to the first power.

In a system of linear equations, we are looking for a point that simultaneously lies on all the lines represented by these equations. This system might have one solution, no solution, or infinitely many solutions. In our exercise, we found only one solution, indicating the lines intersect precisely at one point in space.
  • Each linear equation expresses a relationship between variables.
  • The simplicity of linear equations allows straightforward graphical interpretation.

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Most popular questions from this chapter

Maximize the objective function subject to the constraints \(3 x+y \leq 15,4 x+3 y \leq 30\) \(x \geq 0\), and \(y \geq 0\) $$z=5 x+y$$

Optimal Profit A manufacturer produces two models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2 & 2.5 \\ \hline \text { Painting } & 4 & 1 \\ \hline \text { Packaging } & 1 & 0.75 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 4800 hours, and 1500 hours, respectively. The profits per unit are \(\$ 50\) for model \(\mathrm{A}\) and \(\$ 75\) for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?

Optimal Revenue An accounting firm charges $$\$ 2500$$ for an audit and $$\$ 350$$ for a tax return. Research and available resources have indicated the following constraints. \- The firm has 900 hours of staff time available each week. \- The firm has 155 hours of review time available each week. \- Each audit requires 75 hours of staff time and 10 hours of review time. \- Each tax return requires \(12.5\) hours of staff time and \(2.5\) hours of review time. What numbers of audits and tax returns will bring in an optimal revenue?

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=7 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 8 \end{aligned} $$

Computers The sales \(y\) (in billions of dollars) for Dell Inc. from 1996 to 2005 can be approximated by the linear model \(y=5.07 t-22.4, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Dell Inc.) (a) The total sales during this ten-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 5.07 t-22.4 \\ y \geq 0 \\ t \geq 5.5 \\ t \leq 15.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total sales.
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