Solving Cost Equations: Bread and Milk Pricing

Have you ever faced a problem where you need to determine the cost of individual items based on their grouped prices? In this article, we will walk through a practical example where we need to find the cost of one loaf of bread and one case of milk. This problem is an excellent example of solving a system of linear equations, a fundamental skill in algebra and useful in various real-world scenarios such as pricing analysis.

Problem Statement

Let's assume the cost of two loaves of bread and one case of milk is $35. Similarly, the cost of three loaves of bread and four cases of milk is $69. Our goal is to determine the cost of one loaf of bread and one case of milk.

Setting Up the Equations

We will denote:

b as the cost of one loaf of bread. m as the cost of one case of milk.

From the problem, we can set up the following equations:

2b m 35 3b 4m 69

Solving the System of Equations

To solve this system of equations, we will follow a step-by-step approach:

Step 1: Solve the first equation for m. Step 2: Substitute the expression for m from Step 1 into the second equation. Step 3: Simplify and solve for b. Step 4: Substitute the value of b back into the expression for m.

Step 1: Solve Equation 1 for m

From Equation 1:

m 35 - 2b

Step 2: Substitute m into Equation 2

Substituting m 35 - 2b into Equation 2:

3b 4(35 - 2b) 69

Step 3: Simplify and solve for b

Expanding the equation:

3b 140 - 8b 69

Combine like terms:

-5b 140 69

Subtract 140 from both sides:

-5b 69 - 140

-5b -71

Dividing by -5:

b frac{71}{5} 14.2

Step 4: Substitute b back to find m

Now, substitute b 14.2 back into the expression for m:

m 35 - 2(14.2)

m 35 - 28.4 6.6

Conclusion

The cost of one loaf of bread is $14.20, and the cost of one case of milk is $6.60. This method of solving equations is not only useful in educational settings but also in various real-world applications such as sales and marketing.