Solving Word Problems with Systems of Equations

Word problems can be a fun way to apply your algebra skills to real-world situations. One type of word problem that you might encounter involves setting up and solving a system of equations. Let's dive into an example and see how to tackle it!

The Cookie Jar Problem

Imagine you have a jar of cookies. You want to know how many cookies are in the jar, but you only have some clues.

• If you divide the cookies into groups of 5, you have 2 cookies left over.
• If you divide the cookies into groups of 7, you have 4 cookies left over.

Can you figure out how many cookies are in the jar? Let's use algebra to solve this puzzle!

Setting Up the Equations

Let:

• x represent the total number of cookies

From the clues, we can set up two equations:

• Equation 1: x = 5a + 2 (where a is the number of groups of 5)
• Equation 2: x = 7b + 4 (where b is the number of groups of 7)

Solving the System of Equations

We have two equations with two unknowns (x, a, and b). To solve for x, we can use a method called substitution.

1. Set the two equations equal to each other:

   5a + 2 = 7b + 4

2. Solve for one variable: Let's solve for a:

   5a = 7b + 2

   a = (7b + 2) / 5

3. Substitute the value of a into one of the original equations: Let's use Equation 1:

   x = 5[(7b + 2) / 5] + 2

4. Simplify and solve for x:

   x = 7b + 2 + 2

   x = 7b + 4

We now have an expression for x in terms of b. To find the actual number of cookies, we need to find a value for b that satisfies both original equations. We can do this by trial and error, starting with small values of b.

If we try b = 2, we get:

x = 7(2) + 4 = 18

Let's check if this solution works for both original equations:

• Equation 1: 18 = 5(3) + 2 (True)
• Equation 2: 18 = 7(2) + 4 (True)

Therefore, there are **18 cookies** in the jar.

Key Takeaways

Word problems like this one can be solved by setting up a system of equations and using methods like substitution to find the solution. Remember to carefully define your variables and check your answers to ensure they make sense in the context of the problem.

Practice solving different types of word problems involving systems of equations to improve your understanding and problem-solving skills.