Solving Word Problems with Linear Equations

1. The Strategy: Turning Words into Algebra

The most challenging part of solving a word problem is often the initial translation. Once you have an equation, you already have the skills to solve it. The trick is to develop a systematic approach to the translation process.

Here’s a general framework we'll use:

1. Read and Understand: Read the entire problem to understand the scenario and what you need to find.
2. Define Variables: Identify the unknown quantities and assign variables (like $x$, or more descriptive letters like $c$ for cost) to represent them.
3. Translate and Formulate: Break down the sentences into smaller pieces. Look for keywords and relationships that can be turned into mathematical expressions and, ultimately, one or more equations.
4. Solve: Use the algebraic methods you've learned to solve the equation(s).
5. Answer and Check: State your final answer clearly in a sentence, and check if it makes sense in the context of the original problem.

The most crucial step is number 3. Let's focus on how to translate common English phrases into mathematical symbols.

While the table is a great guide, some phrases, especially for subtraction and division, can be tricky. It's vital to get the order right.

To dive deeper into these nuances, let's watch a video that systematically breaks down phrases for all four basic operations.

2. Problems with a Single Variable

Let's start by applying this translation process to problems that can be modelled with a single linear equation. These problems involve only one unknown quantity, or multiple unknowns that can all be described in terms of one variable.

This next video uses British terminology and works through two clear examples involving age.

Here is another practical example.

3. Problems with Two Variables: Systems of Equations

Now, let's tackle problems that involve two distinct unknown quantities. For these, you'll need to formulate two separate equations—a system of equations—and then use the substitution or elimination methods you learned in the previous lesson.

This is where all our work from this module comes together.

Let's revisit the Maths is Fun resource, as it has excellent examples of this type of problem.

Let's break down the "Sam and Alex" problem to solidify the process:

1. Read and Understand: We need to find how many games Alex played. We know two things: the relationship between the number of games they played, and the total number of games.
2. Define Variables:
   • Let $S$ be the number of games Sam played.
   • Let $A$ be the number of games Alex played.
3. Translate and Formulate:
   • "Sam played 4 more games than Alex" translates to: $S=A+4$
   • "together they played 12 games" translates to: $S+A=12$
4. Solve: We have a system:
   1. $S=A+4$
   2. $S+A=12$
      This is perfectly set up for the substitution method. We can substitute the expression for $S$ from the first equation into the second:
   $$\begin{array}{rl}(A+4)+A& =12\\ 2A+4& =12\\ 2A& =8\\ A& =4\end{array}$$
5. Answer and Check: Alex played 4 games. If Alex played 4, Sam played $4+4=8$. Together they played $4+8=12$. This matches the problem statement. Perfect.

Conclusion

Congratulations on completing the first module! Today, you've learned to connect your algebra skills to the real world by translating word problems into linear equations. This is a crucial skill for applying mathematics in science, technology, and finance.

Key Takeaways:

• Solving word problems is a structured process: Read -> Define -> Translate -> Solve -> Answer.
• Identifying keywords is key to translating English sentences into mathematical operations.
• Simple problems can be modelled with a single equation and one variable.
• More complex problems with two unknowns require a system of two linear equations, which you can solve using substitution or elimination.

This lesson marks the end of our journey into linear equations. We have built a solid foundation, from the order of operations to solving complex word problems.

Preview of the Next Lesson:

In our next module, we will venture beyond straight lines. We'll begin with the topic of indices (exponents). You'll learn the rules for manipulating powers like $x^2$ or $10^5$, including negative and fractional indices. This is the essential groundwork for understanding our next major topic: quadratic equations and polynomial functions.