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Chapter 5: Problem 21

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 3 x-7 y=1 \\ 6 x+5 y=-17 \end{array} $$

Short Answer

Expert verified
The solution to the system of equations is \( x = -2 \) and \( y = -1 \).

Step by step solution

01

Multiply the Equations

To eliminate one of the variables by addition, the coefficients of one variable in both equations must be opposites. Multiply the first equation by 2 so that the coefficients of x in both equations become the same. The first equation becomes: \( 2(3x - 7y) = 2(1) \) which simplifies to: \[ 6x - 14y = 2 \]
02

Add the Equations

Add the modified first equation to the second equation to eliminate the variable x: \( (6x - 14y) + (6x + 5y) = 2 + (-17) \) which simplifies to: \[ 6x + 6x - 14y + 5y = 2 - 17 \] Combine like terms to get: \[ 12x - 9y = -15 \]
03

Solve for y

Isolate y in the obtained equation: \[ 12x - 9y = -15 \] Divide by 3: \[ -3y = -15 \] Divide both sides by -3: \[ y = \frac{-15}{-9} = \frac{15}{9} = \frac{5}{3} \]
04

Substitute y in Original Equation

Substitute \( y = -1 \) into one of the original equations to find x. Using the first original equation: \( 3x - 7 \left( -1 \right) = 1 \) Which simplifies to: \[ 3x + 7 = 1 \] Subtract 7 from both sides: \[ 3x = -6 \] Divide by 3: \[ x = \frac{-6}{3} = -2 \]
05

Verify the Solution

Substitute \( x = -2 \) and \( y = -1 \) back into the second original equation to verify: \( 6 \left( -2 \right) + 5 \left( -1 \right) = -17 \) Simplify: \[ -12 - 5 = -17 \] which holds true. Therefore, the solution is correct.

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Key Concepts

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

System of Equations
A system of equations consists of two or more equations that share the same set of variables. In this exercise, we are dealing with a system of two linear equations involving the variables x and y:

  • \( 3x - 7y = 1 \)
  • \( 6x + 5y = -17 \)
To solve a system of equations, we need to find a common solution that satisfies all the given equations simultaneously. In our case, we aim to find values for x and y that make both equations true. There are several methods to solve systems of equations, including substitution, elimination, and graphing. For this problem, we will use the elimination method, also called the addition method.
Elimination Method
The elimination method, also known as the addition method, involves eliminating one of the variables by adding or subtracting the equations. This helps us to focus on solving for one variable at a time.

  • Step 1: Multiply equations as necessary to match coefficients.
  • Step 2: Add or subtract equations to eliminate one variable.
  • Step 3: Solve for the remaining variable.
  • Step 4: Substitute back to find the other variable.
In this exercise, we multiply the first equation by 2 to make the coefficients of x the same in both equations:
\(2(3x - 7y) = 2(1) \)
which simplifies to:
\[6x - 14y = 2\]
This allows us to add the equations and eliminate x:
\( (6x - 14y) + (6x + 5y) = 2 + (-17) \)
This simplifies to:
\[12x - 9y = -15\]
Proceed by solving for y, and then using that value to solve for x.
Solving Linear Equations
After using the elimination method to simplify the system, we are left with a single linear equation. Solving linear equations involves isolating the variable on one side of the equation to find its value. Let's go step-by-step through the process:

  • Isolate the variable: In the simplified equation \( 12x - 9y = -15 \), isolate y:
  • Divide the whole equation by 3:
    • \[ -3y = -15 \]
  • Divide both sides by -3:
  • Solution for y: \[ y = \frac{15}{9} = \frac{5}{3} \]
Next, substitute this value back into one of the original equations. Using the first equation:
\( 3x - 7 \times \frac{5}{3} = 1 \)

Simplify it to find x, and check the solution in the second original equation to ensure it holds true. Ultimately, you'll find that \(x = -2\) and \(y = -1\), which satisfies both equations in our system, confirming that the solution is correct.

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Most popular questions from this chapter

Two particles begin at the same point and move at different speeds along a circular path of circumference \(280 \mathrm{ft}\). Moving in opposite directions, they pass in \(10 \mathrm{sec} .\) Moving in the same direction, they pass in \(70 \mathrm{sec} .\) Find the speed of each particle.

Solve the system using any method. $$ \begin{array}{l} 2 x-7 y=2400 \\ -4 x+1800=y \end{array} $$

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?

Solve the system using any method. $$ \begin{array}{l} 4(x-2)=6 y+3 \\ \frac{1}{4} x-\frac{3}{8} y=-\frac{1}{2} \end{array} $$

Use the given constraints to find the maximum value of the objective function and the ordered pair \((x, y)\) that produces the maximum value. \(x \geq 0, y \geq 0\) \(x+y \leq 20\) \(x+2 y \leq 36\) \(x \leq 14\) a. Maximize: \(z=12 x+15 y\) b. Maximize: \(z=15 x+12 y\)
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