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Chapter 7: Problem 27

Solving a System of Linear Equations In Exercises \(25 - 46\) , solve the system of linear equations and check any solutions algebraically. $$\left\\{ \begin{aligned} 2 x & \+ 2 z = 2 \\ 5 x + 3 y & = 4 \\ 3 y - 4 z & = 4 \end{aligned} \right.$$

Short Answer

Expert verified
The solution to the system of equations is \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\).

Step by step solution

01

Simplify The First Equation

Notice that the first equation, \(2x + 2z = 2\), can be simplified by dividing through by 2 to give: \(x + z = 1\).
02

Substitute First Equation Into Second Equation

Take the simplified first equation and substitute it into the second equation. This will give: \(5(x + z) + 3y = 4\). This simplifies to \(5 + 3y = 4\), which can further be simplified to give \(3y = -1\). Solve this for \(y\) to give \(y = -1/3\).
03

Substitute \(y\) Into Third Equation

Substitute \(y = -1/3\) into the third equation to get \(3(-1/3) -4z = 4\). Simplify this equation to give \(-4z = 5\) which further simplifies to \(z = -5/4\).
04

Substitute \(z\) Into First Equation

Substitute \(z = -5/4\) into the first simplified equation \(x + z = 1\). Simplify to get \(x = 1 -(-5/4)\), which simplifies to \(x = 9/4\).
05

Verify Solution

Verify the solution by substituting \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\) into the original system of equations. If the values of \(x\), \(y\), and \(z\) satisfy all equations in the original system, the solution is verified.

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Key Concepts

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

Algebraic Substitution Method
The algebraic substitution method is a powerful technique used to solve systems of linear equations. When dealing with multiple variables, finding a solution can seem daunting. However, substitution allows us to reduce the system to one with fewer variables by expressing one variable in terms of another.

Here's the process illustrated: Imagine you have the simplified equation of a system, such as \(x + z = 1\). You can express \(z\) as \(z = 1 - x\) and substitute this expression into another equation in the system. For example, if the second equation is \(5x + 3y = 4\), replacing \(z\) with \(1 - x\) would result in an equation with only \(x\) and \(y\). This method reduces the complexity of the problem and eventually allows you to solve for all variables, one by one.
Simplifying Equations
Simplifying equations is an essential step in solving algebraic problems. It involves reducing the complexity of the equation to its most basic form without changing its solution. For instance, take the equation \(2x + 2z = 2\). This can be simplified by dividing each term by 2, which gives \(x + z = 1\).

Getting into this habit does more than just make the numbers smaller and easier to work with; it also helps to identify the structure of the equation, making substitution clearer. Simplification often includes combining like terms, factoring, and reducing fractions. The goal is to make the next steps of solving the equation, whether that's substitution or another method, as straightforward as possible.
Verifying Solutions Algebraically
Once you believe you have found the solution to a system of equations, it's crucial to verify that the solution is correct. Verifying solutions algebraically involves substituting the values you have found back into the original equations to check if they satisfy every equation in the system.

For example, if you have found solutions \(x = 9/4\), \(y = -1/3\), and \(z = -5/4\), you would replace each variable in the original equations with these values. If each equation balances out, your solution is correct. It's a good habit to always perform this step before considering a problem finished, as it ensures that no errors were made during the problem-solving process.

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Most popular questions from this chapter

Graphical Reasoning Two concentric circles have radii \(x\) and \(y,\) where \(y>x .\) The area between the circles is at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

Solving a Linear Programming Problem, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function (if possible) and where they occur, subject to the indicated constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=5 x+4 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {2 x+2 y \geq 10} \\ {x+2 y \geq 6}\end{array} $$

Defense Department Outlays The table shows the total national outlays \(y\) for defense functions \((\) in billions of dollars) for the years 2004 through 2011 . (Source: \(U . S .\) Office of Management and Budget) (a) Find the least squares regression line \(y=a t+b\) for the data, where \(t\) represents the year with \(t=4\) corresponding to \(2004,\) by solving the system for \(a\) and \(b .\) $$ \left\\{\begin{aligned} 8 b+60 a &=4700.5 \\ 60 b+492 a &=36,865.0 \end{aligned}\right. $$ (b) Use the regression feature of a graphing utility to confirm the result of part (a). (c) Use the linear model to create a table of estimated values of \(y .\) Compare the estimated values with the actual data. (d) Use the linear model to estimate the total national outlay for \(2012 .\) (e) Use the Internet, your school's library, or some other reference source to find the total national outlay for \(2012 .\) How does this value compare with your answer in part (d)? (f) Is the linear model valid for long-term predictions of total national outlays? Explain.

Data Analysis: Wildlife A wildlife management team studied the reproductive rates of deer in three tracts of a wildlife preserve. Each tract contained 5 acres. In each tract, the number of females \(x ,\) and the percent of females \(y\) that had offspring the following year were recorded. The table shows the results.$$ \begin{array} { | c | c | c | c | } \hline \text { Number, } x & { 100 } & { 120 } & { 140 } \\ \hline \text { Percent, y } & { 75 } & { 68 } & { 55 } \\\ \hline \end{array} $$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the percent of females that had offspring when there were 170 females. (d) Use the model to estimate the number of females when 40\(\%\) of the females had offspring.

Optimal Profit A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{X}\) are 3 hours, 3 hours, and 0.8 hour, respectively. The times for model \(Y\) are 4 hours, 2.5 hours, and 0.4 hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(\$ 300\) for model \(X\) and \(\$ 375\) for model \(Y .\) What is the optimal production level for each model? What is the optimal profit?
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