Problem 12 - Elementary and Intermediate Algebra

Question

Telemarketing. Sven, Laurie, and Isaiah can process 740 telephone orders per day. Sven and Laurie together can process 470 orders, while Laurie and Isaiah together can process 520 orders per day. How many orders can each person process alone?

Step-by-Step Solution

Answer

Sven: 220, Laurie: 250, Isaiah: 270

Step 1: Define the Variables

Let Sven, Laurie, and Isaiah be represented by the variables S, L, and I respectively. These variables will represent the number of orders each person can process alone.

Step 2: Set Up Equations

Based on the information provided, we can create the following equations: 1) S + L + I = 7402) S + L = 4703) L + I = 520

Step 3: Solve for One Variable

Subtract equation 2 from equation 1 to eliminate L: (S + L + I) - (S + L) = 740 - 470This simplifies to I = 270. Now we have I = 270.

Step 4: Substitute the Known Value

Using I = 270, substitute into equation 3:L + 270 = 520This simplifies to L = 520 - 270So L = 250.

Step 5: Find the Final Unknown

Using the known values of L and I, substitute into equation 2:S + 250 = 470This simplifies to S = 470 - 250So S = 220.

Step 6: Verify the Solution

Confirm that the values satisfy all the original equations:S + L + I = 220 + 250 + 270 = 740S + L = 220 + 250 = 470L + I = 250 + 270 = 520All equations are satisfied.

Key Concepts

variables

In this exercise, variables are used to represent the unknown quantities. Here, we denote the number of telephone orders processed by each person with variables: Sven is represented by S, Laurie by L, and Isaiah by I. Using variables in math helps us create equations that we can solve. A variable can stand for any number that satisfies the equation. In this case, we are trying to find the values of S, L, and I that match the given conditions.

equation solving

Equation solving is the process of finding the values of the variables that make the equation true. In this problem, we start with three equations based on the information provided:
1) S + L + I = 740
2) S + L = 470
3) L + I = 520.
These equations are interdependent, and solving one can help solve the others. Our goal is to find numbers for S, L, and I that satisfy all three equations. Solving equations often involves techniques like substitution or elimination, which we'll discuss next.

substitution method

The substitution method involves solving one equation for one variable and then substituting that solution into other equations. Here’s how it’s used in this problem:
First, we solve one of the equations involving two of the three variables. For instance, subtract equation 2 from equation 1 to eliminate L:
(S + L + I) - (S + L) = 740 - 470, simplifying to I = 270.
Next, substitute this I value into another equation, like equation 3:
L + 270 = 520, simplifying to L = 250.
Finally, use these found values in another equation to solve for the last variable. For example, substituting L into equation 2:
S + 250 = 470, simplifying to S = 220.
This method works by breaking down the problem step-by-step, making it easier to solve.

elimination method

The elimination method involves combining equations to eliminate one variable, making it easier to solve for the others. In this scenario, we eliminate L by subtracting equation 2 from equation 1:
(S + L + I) - (S + L) = 740 - 470, giving us I = 270.
Now, we use the value of I in other equations to eliminate another variable. For example, substituting I = 270 into equation 3:
L + 270 = 520, which simplifies to L = 250.
Next, substitute L into equation 2:
S + 250 = 470, simplifying to S = 220.
Elimination helps simplify the problem by removing one variable at a time, similar to peeling layers off an onion.