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Chapter 8: Problem 88

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Short Answer

Expert verified
The solution to the system of the given equations is \(x = 6\) and \(y = -4\).

Step by step solution

01

Multiply the Equations

Modifying the given equations to make the coefficients of 'y' in both the equations same which can cancel out each other. Selecting the variable 'y' to eliminate, Multiply the first equation by 3 and the second equation by 5, to make the coefficients of y the same in both equations. So, the equations become: \(9x + 15y = -6\) (equation 1) and \(10x + 15y = 0\) (equation 2)
02

Add the Equations

Add the two equations together. The addition of the two equations eliminates the 'y'. So, the equation now becomes: \(-x = -6\).
03

Solve for the remaining variable

Divide both sides by -1 to solve for 'x'. So, \(x = 6\).
04

Substitute x = 6 into one of the original equations

Select one of the original equations to substitute 'x' value into. For example, the original equation \(2x + 3y = 0\) becomes: \(2(6) + 3y = 0\). Which simplifies to: \(12 + 3y = 0\).
05

Solve for 'y'

Subtract 12 from both sides: \(3y = -12\). Divide both sides by 3 to solve for 'y'. So, \(y = -4\).

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Most popular questions from this chapter

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is 125 for the rear-projection televisions and 200 for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is 600 for the rear-projection televisions and 900 for the plasma televisions. Total monthly costs cannot exceed 360,000. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200)\) \((450,100), \text { and }(450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing _____ rear- projection televisions each month and _____ \(-\) plasma televisions each month. The maximum monthly profit is $_____.

will help you prepare for the material covered in the next section. Solve by the substitution method: $$\left\\{\begin{array}{l}{4 x+3 y=4} \\\\{y=2 x-7}\end{array}\right.$$

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=5 x-2 y\\\ &\left\\{\begin{array}{l} {0 \leq x \leq 5} \\ {0 \leq y \leq 3} \\ {x+y \geq 2} \end{array}\right. \end{aligned} $$

In your own words, describe how to solve a linear programming problem.

What kinds of problems are solved using the linear programming method?
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