Problem 14 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. On Monday John walks for 1 hour, jogs for 2 hours, and covers a total of \(28 \mathrm{km}\). On Tuesday he walks for 2 hours, jogs for 1 hour, and covers a total of \(23 \mathrm{km} .\) What are his rate walking and his rate jogging?

Step-by-Step Solution

Answer

John's walking rate is \( 6 \mathrm{km/h} \) and his jogging rate is \( 11 \mathrm{km/h} \).

Step 1: Define Variables

Let's define the variables. Let \( w \) represent John's walking rate in km/h and \( j \) represent John's jogging rate in km/h.

Step 2: Set Up Equations Using Rates

From the problem, we can set up the following equations based on John's total distance covered each day. For Monday: \[ 1w + 2j = 28 \] For Tuesday: \[ 2w + 1j = 23 \]

Step 3: Solve the System of Equations

To solve the system of equations, we can use either the substitution method or the elimination method. We'll use the elimination method here: First, multiply the first equation by 2: \[ 2w + 4j = 56 \] Now, subtract the second equation from the modified first equation: \[ (2w + 4j) - (2w + 1j) = 56 - 23 \] Then we have: \[ 3j = 33 \] Solve for \( j \): \[ j = 11 \]

Step 4: Solve for the Other Variable

Now substitute \( j = 11 \) back into one of the original equations to find \( w \). Using the first equation: \[ w + 2(11) = 28 \] \[ w + 22 = 28 \] \[ w = 6 \]

Step 5: State the Solution

John's walking rate is \( 6 \mathrm{km/h} \) and his jogging rate is \( 11 \mathrm{km/h} \).

Key Concepts

Systems of Equations

In algebra, a system of equations is a set of two or more equations with the same variables. We usually solve systems to find the values of these variables that satisfy all the equations at once.
When solving a real-life problem like John's walking and jogging rates, we can set up a system of equations based on the given conditions.
For instance, John walks and jogs at different rates on two different days, giving us two equations to work with. Systems of equations can be solved using various methods such as substitution, elimination, or graphing.

Variable Definition

Defining variables is a crucial step when solving word problems in algebra.
Properly labeled variables give us a clear understanding of what each symbol represents, making it easier to set up and solve equations.
In the given problem, we defined two variables:

• \( w \) for John's walking rate in km/h
• \( j \) for John's jogging rate in km/h

This clear definition helps us form mathematical equations from the verbal descriptions. For example, John's Monday activity gives us the equation \( w + 2j = 28 \). Similarly, his Tuesday activity gives us \( 2w + 1j = 23 \).

Elimination Method

The elimination method is one way to solve systems of equations. In this method, we aim to eliminate one variable by adding or subtracting equations, simplifying the system to a single-variable equation.
In our example, we first multiplied the Monday equation by 2: \( 2w + 4j = 56 \).
This creates a new system where one of the variables can be eliminated by subtraction: \((2w + 4j) - (2w + 1j) = 33 \) results in \( 3j = 33 \), giving us \( j = 11 \).
Once we have found one variable, substitution (another method) helps to find the value of the other variable.

Solving Linear Equations

After simplifying our system with the elimination method, we need to solve the resulting linear equations.
In our example, after finding \( j = 11 \), we substitute this back into one of our original equations to find the value of \( w \).
Using \( w + 2(11) = 28 \), we simplify it to \( w + 22 = 28 \). Solving this gives us \( w = 6 \).
Linear equations can be solved using basic algebraic operations like addition, subtraction, multiplication, and division.

• Identify the variable you need to isolate.
• Use inverse operations to move other terms to the opposite side of the equation.

Finally, verify the solution by plugging the values back into the original equations to ensure they hold true.