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Chapter 6: Problem 12

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. How many \(37c\) and \(80c\) stamps did Joe buy if he bought 50 stamps and paid \(\$ 24.52\) for them?

Short Answer

Expert verified
Joe bought 36 of the 37-cent stamps and 14 of the 80-cent stamps.

Step by step solution

01

- Define variables

Let the number of 37-cent stamps be represented by \(x\) and the number of 80-cent stamps be represented by \(y\).
02

- Set up the system of equations

Using the information given, we can create the following equations. The first equation represents the total number of stamps: \[x + y = 50\]. The second equation represents the total cost of the stamps: \[0.37x + 0.80y = 24.52\].
03

- Solve the first equation for one variable

Solve the equation \(x + y = 50\) for \(y\): \[y = 50 - x\].
04

- Substitute into the second equation

Substitute \(y = 50 - x\) into the second equation: \[0.37x + 0.80(50 - x) = 24.52\].
05

- Simplify and solve for \(x\)

Distribute and combine like terms: \[0.37x + 40 - 0.80x = 24.52\]. Simplify: \[-0.43x + 40 = 24.52\]. Subtract 40 from both sides: \[-0.43x = -15.48\]. Solve for \(x\): \[x = \frac{-15.48}{-0.43} = 36\]. So, Joe bought 36 of the 37-cent stamps.
06

- Find the value of \(y\)

Substitute \(x = 36\) into \(y = 50 - x\): \[y = 50 - 36 = 14\]. So, Joe bought 14 of the 80-cent stamps.

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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 consists of multiple equations that share common variables. To solve these, we look for values of the variables that satisfy all equations in the system simultaneously. In the stamp problem, we have two equations: one for the total number of stamps and another for the total cost.
  • Equation 1: The total number of stamps: \( x + y = 50 \).
  • Equation 2: The total cost of the stamps: \( 0.37x + 0.80y = 24.52 \).

These equations form our system. The goal is to find values for \( x \) and \( y \) that make both equations true.
Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into another equation. This way, we can work with a single variable at a time, making the system easier to solve.

In the stamp problem, we solved the first equation for \( y \):
\( y = 50 - x \)
This expression was then substituted into the second equation:
\( 0.37x + 0.80(50 - x) = 24.52 \).
This substitution helped us create a single equation in terms of \( x \), which we then solved step-by-step until we found \( x = 36 \).
Variables in Algebra
Variables are symbols (usually letters) that represent unknown values. In algebra, we use variables to create equations that model real-world situations. In our example:
  • Let \( x \) represent the number of 37-cent stamps.
  • Let \( y \) represent the number of 80-cent stamps.

By defining these variables, we translate the word problem into a system of equations. Once the equations are written, we solve for the variables to discover the unknown values. Here, \( x = 36 \) and \( y = 14 \) indicate that Joe bought 36 37-cent stamps and 14 80-cent stamps, respectively.
Linear Equations
A linear equation is an equation that forms a straight line when graphed on a coordinate plane. It has no exponents higher than one and can be written in the form \( ax + by = c \).

In our problem, both equations are linear:
  • The equation for the total number of stamps: \( x + y = 50 \).
  • The equation for the total cost: \( 0.37x + 0.80y = 24.52 \).

Linear equations are simpler to solve and understand than non-linear equations. By manipulating these equations through methods like substitution, we discover the values of the variables involved.

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Most popular questions from this chapter

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 0.2 x+0.02 y &=8 \\ 4 x-y &=20 \end{aligned}\right.$$

An appliance store offers two lease plans for renting a refrigerator. Plan A costs a Elat fee of \(\$ 75\) plus a monthly rental fee of \(\$ 28 .\) Plan \(B\) costs a flat fee of \(\$ 50\) Slus a monthly rental fee of \(\$ 34\). (A) Write an equation for the total cost \(C\) of renting a refrigerator under both plans for \(m\) months. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(m\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine when each plan is more economical.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Lenore can purchase a car for \(\$ 15,000,\) which will require her to spend an average of \(\$ 80\) per month in repairs and maintenance, or she can lease a car for \(\$ 350\) per month, which includes all repairs and maintenance. After how many months will the leased car and the purchased car cost the same?

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} x+3 y &=5 \\ y &=4 x-7 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A certain mutual fund contains 100 stocks. On a certain day all of the stocks changed price. Three times the number of stocks that went up is 14 more than 8 times the number of stocks that went down. Find how many stocks went up and how many went down.
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