Lesson 4: Applications

If one of the variables in either equation has a coefficient of 1 or -1, what method would you use?

If one of the variables has opposite coefficients in the two equations, would you use elimination using subtraction? Explain.

If you only want to estimate the solution to a system of equations, what method would you use?

A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Lesson Description In this lesson, students learn to translate word problems into systems of two linear equations with two variables. The focus is on the "Translation and Setup" phase, which is often the most challenging for students. Common application types include: Total Value/Quantity Problems: Scenarios like ticket sales or coin problems where one equation represents the count (x + y = total) and the other represents the value (ax + by = total value).Mixture Problems: Combining two different concentrations (e.g., 10% saline and 20% saline) to reach a specific target concentration. Interest/Investment Problems: Splitting a principal amount between two accounts with different interest rates. Break-Even Analysis: Finding the point where a business's cost function equals its revenue function (C(x) = R(x)).
Purpose
The purpose of Section 12.4 is to develop systems thinking. In professional environments, problems rarely involve a single variable; usually, you are trying to balance competing needs or limited resources. By learning to model these scenarios, students develop the ability to organize complex information into a solvable format. This skill is particularly vital for students at Middlesex CC pursuing degrees in Business, Accounting, and the Natural Sciences, where finding "optimal solutions" is a daily task.
Depth of Knowledge (DOK) Level
DOK Level 3
Level 3 (Strategic Thinking & Complex Reasoning): This is the highest level of DOK typically found in Elementary Algebra. Students must analyze a narrative, extract the relevant variables, formulate a system of equations, and choose the most efficient solving method (Substitution vs. Addition). Finally, they must interpret the solution within the original context to ensure it makes sense (e.g., you cannot have a negative number of tickets).

If students have trouble writing the necessary equations for a system in a real-world situation,

Then give them these steps to help them explore, plan, solve, and check.

• Determine the question.

• Describe the variables used for the unknowns.

• Translate the conditions in the problem into two equations.

• Solve the system by the best method.

• Analyze the solution in the context of the situation.

Extension: Have students make up their own real-world problem that can be solved using a system of linear equations. This will help all students understand the concept of solving systems of linear equations.

"you try" on page 284 and practice problems on page 292-295