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Chapter 5: Problem 37

Solve the system using any method. $$ \begin{array}{l} 5(2 x+y)=y-x-8 \\ x-\frac{3}{2} y=\frac{5}{2} \end{array} $$

Short Answer

Expert verified
Simplify and solve the system of equations to find values of x and y using elimination.

Step by step solution

01

- Simplify the first equation

Begin by simplifying the first equation: \[5(2x + y) = y - x - 8\]. Expand and combine like terms: \[10x + 5y = y - x - 8\]. This simplifies to: \[11x + 4y = -8\].
02

- Simplify the second equation

Now, simplify the second equation: \[x - \frac{3}{2}y = \frac{5}{2}\]. Clear the fraction by multiplying everything by 2: \[2x - 3y = 5\].
03

- Use the elimination method

Use the elimination method to solve the system of equations. The equations are: 1) \[11x + 4y = -8\]2) \[2x - 3y = 5\]. First, multiply the second equation by 4 to align the y terms: \[8x - 12y = 20\].
04

- Eliminate y

Add the modified equations to eliminate y: \[11x + 4y = -8\] and \[8x - 12y = 20\]. Adding gives: \[19x - 8y = 12\].
05

- Solve for x

Solve for x in the equation \[19x - 8y = 12\]. To isolate x, use substitution or further simplify the equations.
06

- Substitute back to find y

Once x is found, substitute it back into one of the original equations to find the value of y.

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Key Concepts

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

elimination method
The elimination method is a powerful tool to solve systems of linear equations. This method aims to eliminate one variable by combining equations, letting us solve for the remaining variable. To use it effectively, follow these steps:
  • Align the equations so that adding or subtracting them cancels out one variable.
  • Multiply one or both equations to make the coefficients of one variable opposite in each equation.
For instance, in our given system, we adjust the second equation by multiplying it to match the y terms:
Equation 1: \(11x + 4y = -8\)
Equation 2 after multiplying by 4: \(8x - 12y = 20\).
Adding these equations:
\(11x + 4y + 8x - 12y = -8 + 20\), we simplify to
\(19x - 8y = 12\).
This step eliminates y, simplifying our system to one variable, making it easier to solve.
simplifying equations
Simplifying equations is an essential step when solving systems of linear equations. It involves rewriting equations in a simpler form, making them easier to work with. Here's the breakdown:
  • Distribute any constants through parentheses.
  • Combine like terms on both sides of the equation.
  • Clear fractions by multiplying through by the least common denominator (LCD).
Consider our example: Starting with the first equation \(5(2x + y) = y - x - 8\), we expand to get
\(10x + 5y = y - x - 8\).
By moving all x and y terms to one side, we find:
\(11x + 4y = -8\).
The second equation simplifies by clearing fractions:
\(x - \frac{3}{2}y = \frac{5}{2}\) becomes \(2x - 3y = 5\). Simplifying makes the system more manageable.
substitution method
The substitution method offers another approach to solving systems of linear equations. This technique involves solving one equation for one variable and substituting that expression into the other equation. Here's how it's done:
  • Solve one of the equations for one of the variables.
  • Substitute this expression into the other equation.
  • Solve for the remaining variable.
  • Substitute back to find the other variable.
After simplifying our equations:
Equation 1: \(11x + 4y = -8\)
Equation 2: \(2x - 3y = 5\)
We can solve the second equation for x: \(x = \frac{5 + 3y}{2}\).
Substituting this into the first equation gives us:
\(11(\frac{5 + 3y}{2}) + 4y = -8\),
leading to a single variable equation in \(y\), which simplifies solving the system.

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Most popular questions from this chapter

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is \(\$ 240\) and the profit is \(\$ 160 .\) The cost to build a dining room table is \(\$ 320\) and the profit is \(\$ 240\). (See Examples \(2-3)\) Let \(x\) represent the number of kitchen tables produced per month. Let \(y\) represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling \(x\) kitchen tables and \(y\) dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. \- The number of each type of table cannot be negative. \- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. \- The company can build at most 90 dining room tables. \- The company does not want to exceed a monthly cost of \(\$ 48,000\). c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=\frac{3}{2} x-5\) \(6 x-4 y=20\) a. (2,-2) b. (-4,-11)

A manufacturer produces two models of a gas grill. Grill A requires 1 hr for assembly and \(0.4 \mathrm{hr}\) for packaging. Grill \(B\) requires 1.2 hr for assembly and 0.6 hr for packaging. The production information and profit for each grill are given in the table. (See Example 4\()\) $$ \begin{array}{|l|c|c|c|} \hline & \text { Assembly } & \text { Packaging } & \text { Profit } \\ \hline \text { Grill A } & 1 \mathrm{hr} & 0.4 \mathrm{hr} & \$ 90 \\ \hline \text { Grill B } & 1.2 \mathrm{hr} & 0.6 \mathrm{hr} & \$ 120 \\ \hline \end{array} $$ The manufacturer has \(1200 \mathrm{hr}\) of labor available for assembly and \(540 \mathrm{hr}\) of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $$\$ 110$$ and the profit on grill \(\underline{B}\) units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit?

An investment grows exponentially under continuous compounding. After 2 yr, the amount in the account is \$7328.70. After 5 yr, the amount in the account is \$8774.10. Use the model \(A(t)=P e^{r t}\) to a. Find the interest rate \(r\). Round to the nearest percent. b. Find the original principal \(P\). Round to the nearest dollar. c. Determine the amount of time required for the account to reach a value of \(\$ 15,000 .\) Round to the nearest year.

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).
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