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Chapter 5: Problem 20

$$ \begin{array}{l} 2(x+y)=2-y \\ 4 x-1=2-5 y \end{array} $$$$ \begin{array}{l} 5(x+y)=9+2 y \\ 6 y-2=10-7 x \end{array} $$

Short Answer

Expert verified
First system solution: \( x = -1/2 \), \( y = 1 \). Second system solution: \( x = 2 \), \( y = -1/3 \)

Step by step solution

01

- Analyze and Simplify Each System of Equations

We have two separate systems of equations. We'll start by solving each system individually. The first system is: 1. \(2(x+y) = 2 - y\) 2. \(4x - 1 = 2 - 5y\). The second system is: 1. \(5(x+y) = 9 + 2y\) 2. \(6y - 2 = 10 - 7x\). Let's begin with the first system.
02

- Simplify the First Equation of the First System

The first equation of the first system is \(2(x + y) = 2 - y\). Distribute 2 on the left side: \(2x + 2y = 2 - y\). Now, add \(y\) to both sides: \(2x + 3y = 2\). This is our simplified first equation: (1) \(2x + 3y = 2\)
03

- Simplify the Second Equation of the First System

The second equation is \(4x - 1 = 2 - 5y\). Add 1 to both sides: \(4x = 3 - 5y\). Rearrange to match variable order: \(4x + 5y = 3\). This is our simplified second equation: (2) \(4x + 5y = 3\)
04

- Solve the Simplified First System of Equations

We now solve the two simplified equations (1) \(2x + 3y = 2\) and (2) \(4x + 5y = 3\). Use the substitution or elimination method. Let's use elimination: Multiply the first equation by 2: \(2(2x + 3y = 2)\) which gives \(4x + 6y = 4\). Subtract equation (2) from the new equation: \((4x + 6y) - (4x + 5y) = 4 - 3\) Simplifying, we get: \( y = 1\). Now substitute \( y = 1\) back into equation (1): \(2x + 3(1) = 2\) which simplifies to \(2x + 3 = 2\). Therefore, \(2x = -1\) \(x = -1/2\).
05

- Simplify the First Equation of the Second System

The first equation of the second system is \(5(x + y) = 9 + 2y\). Distribute the 5 on the left side: \(5x + 5y = 9 + 2y\). Subtract \(2y\) from both sides: \(5x + 3y = 9\). This is our simplified first equation: (1) \(5x + 3y = 9\)
06

- Simplify the Second Equation of the Second System

The second equation is \(6y - 2 = 10 - 7x\). Add 2 to both sides: \(6y = 12 - 7x\). Rearrange to match variable order: \(7x + 6y = 12\). This is our simplified second equation: (2) \(7x + 6y = 12\)
07

- Solve the Simplified Second System of Equations

We now solve the two simplified equations (1) \(5x + 3y = 9\) and (2) \(7x + 6y = 12\). Use the substitution or elimination method. Let's use elimination: Multiply the first equation by 2: \(2(5x + 3y = 9)\) which gives \(10x + 6y = 18\). Subtract equation (2) from the new equation: \((10x + 6y) - (7x + 6y) = 18 - 12\) Simplifying, we get: \(3x = 6\) \(x = 2\). Now substitute \(x = 2\) back into equation (1): \(5(2) + 3y = 9\) \(10 + 3y = 9\) which simplifies to \(3y = -1\), \(y = -1/3\).

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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 equations of the first degree, meaning they involve only the first power of the variable. In standard form, a linear equation looks like this: \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. They can often be visualized as straight lines when graphed on a coordinate plane. For example, our given equations \(2(x+y)=2-y\) and \(4x-1=2-5y\), can be rearranged into linear form to better understand their relationships. Linear equations are fundamental in algebra and often serve as the basis for more complex mathematical operations and problem-solving.
Elimination Method
The elimination method is a technique used for solving systems of linear equations. By adding or subtracting equations, one of the variables gets eliminated, simplifying the solution process. Here鈥檚 a step-by-step breakdown on how we used this method for our equations:
  • We started with \(2x + 3y = 2\) and \(4x + 5y = 3\).

  • To eliminate \(x\), we multiplied the first equation by 2 to get \(4x + 6y = 4\).

  • We then subtracted the second equation from this new one: \(4x + 6y - 4x - 5y = 4 - 3\), which simplifies to \(y = 1\).

This effectively removes \(x\) from the equation, making it easier to solve for \(y\). This method is particularly useful when dealing with two-variable systems.
Substitution Method
The substitution method is another technique for solving systems of linear equations. It involves solving one of the equations for one variable and then substituting that solution into the other equation. Here鈥檚 how it works in practice:
  • First, solve one of the equations for one of the variables. For instance, from \(2x + 3y = 2\), we can express \(x\) in terms of \(y\).

  • After finding \(y = 1\), substitute this value back into the original equation \(2x + 3(1) = 2\).

  • This gives \(2x + 3 = 2\), simplifying further to find that \(x = -1/2\).

By substituting the value of one variable into the other equation, we can then find the value of the remaining variable. This method is often straightforward, especially if one of the variables is already isolated.
Algebraic Simplification
Algebraic simplification is the process of reducing expressions or equations into their simplest form by performing operations such as distribution, combining like terms, and factoring.
For example, consider the equation \(5(x + y) = 9 + 2y\).


  • We distribute the 5 to both \(x\) and \(y\) to get \(5x + 5y = 9 + 2y\).

  • Next, we combine like terms by subtracting \(2y\) from both sides: \(5x + 5y - 2y = 9\).

    • This final simplified form \(5x + 3y = 9\) makes the equation easier to solve.
    Algebraic simplification is essential for uncovering the fundamental structure of an equation, aiding in both understanding and solving complex mathematical problems.

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    Most popular questions from this chapter

    Juan borrows \(\$ 100,000\) to pay for medical school. He borrows part of the money from the school whereby he will pay \(4.5 \%\) simple interest. He borrows the rest of the money through a government loan that will charge him \(6 \%\) interest. In both cases, he is not required to pay off the principal or interest during his 4 yr of medical school. However, at the end of \(4 \mathrm{yr}\), he will owe a total of \(\$ 19,200\) for the interest from both loans. How much did he borrow from each source?

    Refer to Section 2.5 for a review of linear cost functions and linear revenue functions. A cleaning company charges \(\$ 100\) for each office it cleans. The fixed monthly cost of \(\$ 480\) for the company includes telephone service and the depreciation on cleaning equipment and a van. The variable cost is \(\$ 52\) per office and includes labor, gasoline, and cleaning supplies. a. Write a linear cost function representing the cost \(C(x)\) (in \(\$$ ) to clean \)x\( offices per month. b. Write a linear revenue function representing the revenue \)R(x)\( (in \$) for cleaning \)x$ offices per month. c. Determine the number of offices to be cleaned per month for the company to break even. d. If 28 offices are cleaned, will the company make money or lose money?

    Solve the system of equations by using the substitution method. (See Example 2\()\) $$ \begin{array}{l} 3 x=2 y-11 \\ 6+5 x=1 \end{array} $$

    A couple has \(\$ 60,000\) to invest for retirement. They plan to put \(x\) dollars in stocks and \(y\) dollars in bonds. For parts (a)-(d), write an inequality to represent the given statement. a. The total amount invested is at most \(\$ 60,000\). b. The couple considers stocks a riskier investment, so they want to invest at least twice as much in bonds as in stocks. c. The amount invested in stocks cannot be negative. d. The amount invested in bonds cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).

    A large department store needs at least 3600 labor hours covered per week. It employs full-time staff \(40 \mathrm{hr} / \mathrm{wk}\) and part-time staff \(25 \mathrm{hr} / \mathrm{wk}\). The cost to employ a full-time staff member is more because the company pays benefits such as health care and life insurance. $$ \begin{array}{|l|c|c|} \hline & \text { Hours per Week } & \text { Cost per Hour } \\ \hline \text { Full time } & 40 \mathrm{hr} & \$ 20 \\ \hline \text { Part time } & 25 \mathrm{hr} & \$ 12 \\ \hline \end{array} $$ The store manager also knows that to make the store run efficiently, the number of full-time employees must be at least 1.25 times the number of part- time employees. a. Determine the number of full-time employees and the number of part-time employees that should be used to minimize the weekly labor cost. b. What is the minimum weekly cost to staff the store under these constraints?
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