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Chapter 7: Problem 110

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.

Short Answer

Expert verified
1. An apple and an orange each cost $2 and $3 respectively. 2. One friendship group has 19 members, and the other one has 16 members. 3. There are 40 novels and 70 dictionaries in the library. 4. The recreation center has 60 adult members and 60 child members.

Step by step solution

01

Problem 1 - Grocery Store Expenses

Suppose a customer bought 5 apples and 7 oranges from a grocery store for $30. On the following day, the customer bought 3 apples and 2 oranges for $14. These two purchases can be represented as two mathematical equations that form a system of equations. Let \(X\) represent the price of an apple and \(Y\) the price of an orange. Then the equations are: \(5X + 7Y = 30\) and \(3X + 2Y = 14\). Solving this system yields \(X = 2\) and \(Y = 3\). Hence, an apple costs $2 and an orange costs $3.
02

Problem 2 - School Friendship Groups

In a school, there are 35 students in two friendship groups. One group has 4 more members than the other. This information sets up a system of equations where \(X\) and \(Y\) represent the number of members in the two groups: \(X + Y = 35\) and \(X = Y + 4\). Solving this system indicates that one group has 19 members and the other has 16 members.
03

Problem 3 - Library Books

A school library has 110 books, which are either novels or dictionaries. The number of dictionaries is 30 more than the number of novels. Let \(X\) be the number of novels and \(Y\) the number of dictionaries. The two equations are: \(X + Y = 110\) and \(Y = X + 30\). Solving this system indicates that there are 40 novels and 70 dictionaries in the library.
04

Problem 4 - Recreation Center Membership

A recreation center has 120 members, consisting of adults and children. Adult membership fees are $25, and child membership fees are $10. The total membership income is $2100. If we let \(X\) be the number of adult members and \(Y\) the number of child members, we get the system: \(X + Y = 120\) and \(25X + 10Y = 2100\). By solving this system, it's revealed that there are 60 adults and 60 children as members.

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Key Concepts

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

Understanding Word Problems in Systems of Linear Equations
Word problems are a core element in mathematics that translate real-life situations into mathematical language. They鈥檙e essential for practicing critical thinking and problem-solving skills. When word problems involve systems of linear equations, they often describe scenarios with two unknowns, which require setting up and solving equations.
  • Identify what each variable represents in the context of the problem.
  • Translate the given information into mathematical equations, using the variables appropriately.
  • Consider what operations or formations are necessary to describe the relationships.
Breaking down the story into equations makes complex problems easier to manage. Step-by-step translation from words to equations is crucial in finding the solution.
How to Solve Linear Equations from Word Problems
Solving linear equations involves finding the unknown variables that satisfy all given equations. The process typically includes several steps:
  • First, simplify each equation if possible. Combine like terms and eliminate fractions when you can.
  • Next, look to isolate one variable. This can be done using substitution or elimination methods.
  • Then, solve for the first variable, and substitute back to find the second variable.
  • Finally, verify your solution by plugging the values back into the original equations to ensure they are correct.
By following these steps, word problems that initially seem complicated become manageable. Consistently practicing with different scenarios helps to strengthen problem-solving skills.
The Role of Two Variables in System of Equations
In many word problems, two variables represent different unknown quantities that are related. For problems involving systems of linear equations, distinct variables like \(X\) and \(Y\) often describe these unknowns.
  • Each variable corresponds to a real-world element, such as the price of items or the number of people.
  • The relationships between the variables are expressed through linear equations based on the given conditions.
  • Balancing these equations allows you to uncover the specific values of the variables.
Effectively setting up the right variables and equations is crucial in solving the system. This step ensures that every part of the problem is accurately represented in the mathematical model.
Exploring Algebraic Solutions in Linear Equations
Algebraic solutions are powerful tools in solving systems of linear equations. They focus on utilizing algebraic methods like substitution and elimination to derive solutions.
  • Substitution involves solving one equation for one variable and then substituting that expression into another equation.
  • Elimination requires adding or subtracting equations to eliminate a variable, making it easier to solve for the remaining one.
  • These methods transform potentially complex problems into simpler, solvable parts.
The key to mastering algebraic solutions is practice and familiarity with manipulating equations. Over time, these techniques can be applied quickly and confidently to a variety of word problems.

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Most popular questions from this chapter

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+x y=6 \\ x^{2}+2 x y=0 \end{array}\right.$$

Use a system of linear equations to solve Exercises. The current generation of college students grew up playing interactive online games, and many continue to play in college. The bar graph shows the percentage of U.S. college students playing online games, by gender. (GRAPH CAN'T COPY) A total of \(41 \%\) of college men play online games multiple times per day or once per day. The difference in the percentage who play multiple times per day and once per day is \(7 \% .\) Find the percentage of college men who play online games multiple times per day and the percentage of college men who play online games once per day.

Bottled water and medical supplies are to be shipped to survivors of an earthquake by plane. The bottled water weighs 20 pounds per container and medical kits weigh 10 pounds per kit. Each plane can carry no more than \(80,000\) pounds. If \(x\) represents the number of bottles of water to be shipped per plane and \(y\) represents the number of medical kits per plane, write an inequality that models each plane's \(80,000\) -pound weight restriction.

Find the domain of each function. $$g(x)=\frac{x-6}{x^{2}-36}$$

Use a system of linear equations to solve Exercises. A rectangular lot whose perimeter is 320 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 16\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 5\) per foot. The total cost of the fencing along the three sides comes to \(\$ 2140 .\) What are the lot's dimensions?
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