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Chapter 3: Problem 49

Anna purchased 32 strings for her autoharp. Wrapped strings cost \(\$ 4.49\) each and unwrapped strings cost \(\$ 2.99\) each. If she paid a total of \(\$ 107.68\) for the strings, how many of each type did she buy?

Short Answer

Expert verified
Anna bought 8 wrapped strings and 24 unwrapped strings.

Step by step solution

01

- Define the Variables

Let the number of wrapped strings be denoted as \( w \) and the number of unwrapped strings be denoted as \( u \).
02

- Set Up the Equations

We know the total number of strings is 32. Therefore, we can write the equation: \[ w + u = 32 \]Additionally, we know the total cost of the strings is \$107.68, with wrapped strings costing \$4.49 each and unwrapped strings costing \$2.99 each. This gives us the second equation: \[ 4.49w + 2.99u = 107.68 \]
03

- Solve the System of Equations

First, solve the first equation for one of the variables (for example, \( u \)): \[ u = 32 - w \]Substitute this expression for \( u \) into the second equation: \[ 4.49w + 2.99(32 - w) = 107.68 \]Simplify and solve for \( w \):\[ 4.49w + 2.99 \times 32 - 2.99w = 107.68 \]\[ 4.49w + 95.68 - 2.99w = 107.68 \]\[ 1.5w + 95.68 = 107.68 \]\[ 1.5w = 12 \]\[ w = \frac{12}{1.5} \]\[ w = 8 \]
04

- Find the Value of the Remaining Variable

Substitute the value of \( w \) back into the equation \( u = 32 - w \):\[ u = 32 - 8 \]\[ u = 24 \]
05

- Verify the Solution

Check the values by substituting them back into the original equation to ensure they satisfy both equations:First equation: \[ 8 + 24 = 32 \]Second equation: \[ 4.49 \times 8 + 2.99 \times 24 = 107.68 \]\[ 35.92 + 71.76 = 107.68 \] Both equations are satisfied.

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Key Concepts

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

Variables
In solving a system of linear equations, we start by defining the variables. Variables represent the unknown quantities we need to find.
In our problem, we have two types of strings: wrapped and unwrapped.
We use variables to denote these quantities:
  • Let w signify the number of wrapped strings.
  • Let u denote the number of unwrapped strings.
Choosing appropriate variables helps in formulating the equations correctly and solving the problem systematically.
Linear Equations
Linear equations are mathematical statements of equality involving variables, constants, and linear terms.
In our exercise:
  • The first linear equation, based on the total number of strings, is: \( w + u = 32 \).
  • The second linear equation, based on the total cost, is: \( 4.49w + 2.99u = 107.68 \).
Both equations are linear because the variables w and u are raised to the power of 1.
These equations help us represent the problem mathematically.
Substitution Method
The substitution method is a technique for solving a system of linear equations. It involves isolating one variable in one equation and substituting this expression into the other equation.
In our solution, we:
  • Resolved the first equation for u: \( u = 32 - w \).
  • Substituted this value into the second equation to get one equation in terms of w only: \( 4.49w + 2.99(32 - w) = 107.68 \).
This simplification helps us find the values for both variables step by step.
Verification of Solutions
Verification ensures that the solutions obtained actually satisfy both original equations.
To verify, substitute the found values back into the given equations:
  • For the first equation: \( 8 + 24 = 32 \), which is true.
  • For the second equation: \( 4.49 \times 8 + 2.99 \times 24 = 107.68 \), which also holds true.
Both equations are satisfied with the values w = 8 and u = 24.
The solution's correctness is confirmed when both checks pass.
Problem-Solving Steps
Approaching a problem with a clear plan simplifies finding the solution. Our systematic steps include:
  • Define Variables: Choose variables to represent the unknowns.
  • Set Up Equations: Formulate the problem using linear equations based on given information.
  • Solve the System: Use methods like substitution to find the values of the variables.
  • Verify the Solution: Check the found values in the original equations to confirm correctness.
Following these steps ensures a structured and efficient problem-solving process.
It helps in tackling not just this problem but many other similar problems as well.

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Most popular questions from this chapter

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} u-v+6 w &=8 \\ 3 u-v+6 w &=14 \\ -u-2 v-3 w &=7 \end{aligned} $$

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