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Chapter 5: Problem 26

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 0.25 x-0.04 y=0.24 \\ 0.15 x-0.12 y=0.12 \end{array} $$

Short Answer

Expert verified
x = 1, y = 0.25

Step by step solution

01

Align the Equations

Write the system of equations aligned and clear for elimination method: \[ \begin{array}{l} 0.25x - 0.04y = 0.24 \ 0.15x - 0.12y = 0.12 darray \] Focus on creating a situation where we can eliminate one variable.
02

Equalize Coefficients

Find a common multiplier for the coefficients of either x or y so you can eliminate one variable. Here, we'll focus on eliminating y. For that, multiply the first equation by 3 and the second equation by 1: \[ \begin{array}{l} 3(0.25x - 0.04y) = 3(0.24) \ 1(0.15x - 0.12y) = 1(0.12) darray \] which transforms the system into: \[ \begin{array}{l} 0.75x - 0.12y = 0.72 \ 0.15x - 0.12y = 0.12 darray \]
03

Subtract the Equations

Subtract the second equation from the first to eliminate y: \[ (0.75x - 0.12y) - (0.15x - 0.12y) = 0.72 - 0.12 darray \] Simplify this to: \[ 0.60x = 0.60 darray \]
04

Solve for x

Solve the simplified equation: \[ x = \frac{0.60}{0.60} = 1 darray \]
05

Solve for y

Substitute x = 1 back into one of the original equations to solve for y. Using the first equation: \[ 0.25(1) - 0.04y = 0.24 darray \] Simplify to: \[ 0.25 - 0.04y = 0.24 darray \] then isolate y: \[ 0.04y = 0.25 - 0.24 = 0.01 darray \] so, \[ y = \frac{0.01}{0.04} = 0.25 darray \]

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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 is a set of two or more equations with the same variables. In our example, we have two equations: \(0.25x - 0.04y = 0.24\) and \(0.15x - 0.12y = 0.12\). To solve these equations, we are looking for values of \(x\) and \(y\) that satisfy both equations simultaneously. This means that both equations are true at the same time for those \(x\) and \(y\) values.
Systems of equations can be solved using various methods, such as graphing, substitution, or the addition method, also known as the elimination method. The choice of method depends on the particular system and which approach seems most straightforward.
elimination method
The elimination method is a technique for solving a system of equations by removing one of the variables. This is done by adding or subtracting the equations. The goal is to form a new equation that contains only one variable.

To achieve this elimination:
  • Identify which variable to eliminate. In our example, it's easier to eliminate \(y\) since the coefficients can be adjusted to match relatively simply.
  • Multiply the equations by certain numbers to make the coefficients of the chosen variable (in our case, \(y\)) equal but opposite in sign.
  • Subtract one equation from the other, effectively canceling out the chosen variable.


    • For example, in our exercise, we multiplied the first equation by 3 to make the coefficients of \(y\) matching (both become \(-0.12y\). After this, subtracting one equation from the other eliminates \(y\), leaving an equation with just \(x\).
solving equations step by step
Solving equations step by step is a logical and sequential approach to find the values of variables. Let's break down our given system into clear steps:
  • Align the equations: Make sure they're lined up neatly for an orderly work process.
  • Equalize coefficients: Adjust the equations so that the coefficients of one variable are equal, making it easier to eliminate that variable.
  • Subtract or add equations: This will eliminate one variable and simplify the system to a single-variable equation.
  • Solve for one variable: With the single-variable equation, solve to find the value of that variable.
  • Substitute back: Plug the value obtained into one of the original equations to solve for the remaining variable.
In our example, we followed these steps, resulting in \(x = 1\) and then used that to find \(y = 0.25\). Taking it one step at a time ensures clarity and accuracy.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations using algebraic properties to isolate the variables. This can include:
  • Adding or subtracting terms from both sides of an equation.

  • Multiplying or dividing both sides by a nonzero number.

  • Using the distributive property to expand or factor expressions.

In the given solution, algebraic manipulation was used in several steps:
  • Multiplying the equations to create equal coefficients for \(y\).

  • Subtracting equations to eliminate \(y\) and simplify to a single-variable equation.

  • Simplifying to isolate \(x\) and \(y\) by performing basic arithmetic operations.

Mastering these skills is crucial for solving not only systems of equations but also a wide range of algebraic problems.

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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?

Jonas performed an experiment for his science fair project. He learned that rinsing lettuce in vinegar kills more bacteria than rinsing with water or with a popular commercial product. As a follow-up to his project, he wants to determine the percentage of bacteria killed by rinsing with a diluted solution of vinegar. a. How much water and how much vinegar should be mixed to produce 10 cups of a mixture that is \(40 \%\) vinegar? b. How much pure vinegar and how much \(40 \%\) vinegar solution should be mixed to produce 10 cups of a mixture that is \(60 \%\) vinegar?

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} 3 x-4 y=6 \\ 9 x=12 y+4 \end{array} $$

To protect soil from erosion, some farmers plant winter cover crops such as winter wheat and rye. In addition to conserving soil, cover crops often increase crop yields in the row crops that follow in spring and summer. Suppose that a farmer has 800 acres of land and plans to plant winter wheat and rye. The input cost for 1 acre for each crop is given in the table along with the cost for machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Wheat } & \$ 90 & \$ 50 & \$ 42 \\ \hline \text { Rye } & \$ 120 & \$ 40 & \$ 35 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 90,000$$ for input costs and a maximum of $$\$ 36,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre for wheat were $$\$ 40$$ and the profit per acre for rye were $$\$ 45$$, how many acres of each crop should be planted to maximize profit?

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