Solving for Joan and Mary's Money: A Simple Algebraic Equation

When dealing with algebraic equations, one of the simplest and most straightforward problems to solve can involve determining the amount of money two individuals have based on the relationships between their possessions. This particular problem involves Joan and Mary, and we will use algebra to solve it step by step.

Problem Statement

Joan has 5 more than Mary. Together they have 40. How much does each have?

Solving the Problem

Step 1: Define the Variables

Let's denote the amount of money Mary has as M. Since Joan has 5 more than Mary, we can express Joan's amount as M 5. Combined, they have 40, so the equation is:

M M 5 40

Step 2: Simplify the Equation

We can simplify the equation as follows:

2M 5 40

Step 3: Isolate the Variable M

Subtract 5 from both sides of the equation to isolate the term with M:

2M 35

Divide both sides by 2 to solve for M:

M 17.50

Step 4: Calculate Joan's Amount

Since Joan has 5 more than Mary, we calculate:

Joan's amount M 5 17.50 5 22.50

So, Mary has $17.50 and Joan has $22.50.

Proving the Solution

Proof by Equations

Let's verify our solution by substituting the values back into the original equations:

Equation 1: M (M 5) 40

17.50 (17.50 5) 40

17.50 22.50 40

40 40 nbsp; (This is true)

Equation 2: (M 5) - M 5

22.50 - 17.50 5

5 5 nbsp; (This is true)

Algebraic Solution Explained

In this algebraic solution, we used the following steps:

Defined the variables. Formulated the equation based on the given conditions. Simplified the equation to isolate the variable. Solved for the variable and calculated the other value. Verified the solution using the original conditions.

Conclusion

By solving this simple algebraic problem, we determined that Mary has $17.50 and Joan has $22.50. This exercise demonstrates the power of algebraic equations in solving real-world problems involving variables and relationships.

Keywords: algebraic equations, simultaneous equations, solving for variables