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Chapter 7: Problem 55

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $$\$ 82$$. She forgot to save the individual receipts but remembered that your groceries were $$\$ 0.05$$ cheaper than half of her groceries, and that your other roommate's groceries were $$\$ 2.10$$ more than your groceries. How much was each of your share of the groceries?

Short Answer

Expert verified
Sarah: $40, You: $19.95, Other roommate: $22.05.

Step by step solution

01

Define Variables

Let Sarah's groceries cost \( S \), your groceries cost \( Y \), and your other roommate's groceries cost \( O \).
02

Set Up Equations

Using the information given, we can set up the equations: - Total bill: \( S + Y + O = 82 \)- Your groceries: \( Y = \frac{1}{2}S - 0.05 \)- Other roommate's groceries: \( O = Y + 2.10 \).
03

Substitute for Y and O

Substitute the expressions for \( Y \) and \( O \) in terms of \( S \) from the second step into the first equation: \( S + \frac{1}{2}S - 0.05 + \left( \frac{1}{2}S - 0.05 + 2.10 \right) = 82 \).
04

Simplify the Equation

Simplify the equation: \( S + \frac{1}{2}S - 0.05 + \frac{1}{2}S - 0.05 + 2.10 = 82 \). Combine like terms: \( 2S + 2.00 = 82 \).
05

Solve for S

Rearrange the equation to solve for \( S \):\( 2S = 80 \) Thus, \( S = 40 \).
06

Find Your Groceries (Y)

Using \( S = 40 \) in the equation for \( Y \):\( Y = \frac{1}{2}(40) - 0.05 = 20 - 0.05 = 19.95 \).
07

Find Other Roommate's Groceries (O)

Using \( Y = 19.95 \) in the equation for \( O \):\( O = 19.95 + 2.10 = 22.05 \).

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Key Concepts

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

Defining Variables
Whenever you encounter a problem that involves unknown quantities, it's crucial to clearly define your variables.This helps simplify and organize the information in the problem. In our example, the task is to find the cost of groceries for each roommate.
You will use variables to represent these costs:
  • Let \( S \) represent the cost of Sarah's groceries.
  • Let \( Y \) represent the cost of your groceries.
  • Let \( O \) represent the cost of your other roommate's groceries.
By assigning these variables, we make it easier to see the relationships between different parts of the problem. Being consistent with variable names is key to minimizing confusion as you proceed with solving the exercise.
Setting Up Equations
After defining the variables, the next step is to translate the word problem into mathematical equations.This involves using the relationships described in the problem.
  • The total cost of groceries was known: \( S + Y + O = 82 \).
  • Your groceries were \( \\(0.05 \) cheaper than half of Sarah鈥檚 groceries: \( Y = \frac{1}{2}S - 0.05 \).
  • The other roommate鈥檚 groceries cost \\)2.10 more than yours: \( O = Y + 2.10 \).
Turning relationships into equations is like creating a blueprint for solving the problem.Everything you need to find out is now mathematically represented.
Substitution Method
The substitution method is a technique used to solve systems of equations.After setting up your equations, you can use substitution to solve for one of the variables.Start by using one equation to express one variable in terms of another.In this problem, use the equations for \( Y \) and \( O \) to substitute into the total cost equation.Replace \( Y \) and \( O \) in the total cost equation:\[S + \frac{1}{2}S - 0.05 + \left(\frac{1}{2}S - 0.05 + 2.10\right) = 82\]This allows you to reduce the number of variables in one equation, making it easier to solve.
Simplification
Simplification in algebra involves combining like terms and clearing out any unnecessary elements from an equation.It's an important step to make solving the equations easier and more straightforward.In this problem, simplify the substitution equation from:\[S + \frac{1}{2}S - 0.05 + \frac{1}{2}S - 0.05 + 2.10 = 82\]Combine like terms:\[2S + 2.00 = 82\]Simplification helps to see the direct relationships and to better manipulate the equation to find the solution.
Solving Linear Equations
The final step is to solve the simplified linear equation for the variable of interest. Once simplified, our equation is: \[2S + 2.00 = 82\]The next move is to isolate \( S \).Subtract \( 2.00 \) from both sides to get:\[2S = 80\]Divide both sides by \( 2 \) to solve for \( S \):\[S = 40\]Once \( S \) is found, you can use it to find other variable values:
  • For \( Y \), substitute \( S = 40 \) into \( Y = \frac{1}{2}(40) - 0.05 = 19.95 \).
  • For \( O \), use \( Y = 19.95 \) in \( O = 19.95 + 2.10 = 22.05 \).
By following these steps, each variable's value is revealed, solving the whole problem.

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Most popular questions from this chapter

For the following exercises, use the determinant function on a graphing utility. \(\left|\begin{array}{rrrr}\frac{1}{2} & 1 & 7 & 4 \\ 0 & \frac{1}{2} & 100 & 5 \\ 0 & 0 & 2 & 2,000 \\ 0 & 0 & 0 & 2\end{array}\right|\)

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 5 x+2 y-z &=1 \\ -7 x-8 y+3 z &=1.5 \\ 6 x-12 y+z &=7 \end{aligned} $$

For the following exercises, solve a system using the inverse of a \(3 \times 3\) matrix. $$\begin{aligned} \frac{1}{2} x-\frac{1}{5} y+\frac{1}{5} z &=\frac{31}{100} \\\\-\frac{3}{4} x-\frac{1}{4} y+\frac{1}{2} z &=\frac{7}{40} \\\\-\frac{4}{5} x-\frac{1}{2} y+\frac{3}{2} z &=14 \end{aligned}$$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. Men aged \(20-29,30-39\) , and \(40-49\) made up 78\(\%\) of the population at a prison last year. This year, the same age groups made up 82.08\(\%\) of the population. The \(20-29\) age group increased by \(20 \%,\) the \(30-39\) age group increased by \(2 \%,\) and the \(40-49\) age group decreased to \(\frac{3}{4}\) of their previous population. Originally, the \(30-39\) age group had 2\(\%\) more prisoners than the \(20-29\) age group. Determine the prison population percentage for each age group last year.

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{aligned} 4 x-3 y+4 z &=10 \\ 5 x-2 z &=-2 \\ 3 x+2 y-5 z &=-9 \end{aligned} $$
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