Show me an example problem for SAT [SAT Math]

Alright, let’s tackle a classic SAT math problem! This example will focus on algebra, a core component of the SAT. We’ll break down the problem step-by-step, highlighting key concepts and strategies for success.

### SAT Algebra Example Problem: Linear Equations and Systems

**Problem:**

A landscaping company charges a flat fee of $50 for a consultation, plus $35 per hour for labor. Another landscaping company charges a flat fee of $75 for a consultation, plus $30 per hour for labor. For what number of hours of labor will the total cost be the same for both companies?

#### Step-by-Step Solution

1. **Define Variables:**

* Let *h* represent the number of hours of labor.
* Let *C* represent the total cost.

2. **Write Equations:**

* Company 1: $C = 50 + 35h$
* Company 2: $C = 75 + 30h$

3. **Set the Equations Equal:**

Since we want to find when the total cost is the same for both companies, we set the two equations equal to each other:

$50 + 35h = 75 + 30h$

4. **Solve for *h*:**

* Subtract 30*h* from both sides:

$50 + 35h – 30h = 75 + 30h – 30h$

$50 + 5h = 75$

* Subtract 50 from both sides:

$50 + 5h – 50 = 75 – 50$

$5h = 25$

* Divide both sides by 5:

$frac{5h}{5} = frac{25}{5}$

$h = 5$

5. **Answer:**

The total cost will be the same for both companies when the labor is for 5 hours.

#### Verification

Let’s verify our answer by plugging *h* = 5 into both equations:

* Company 1: $C = 50 + 35(5) = 50 + 175 = 225$
* Company 2: $C = 75 + 30(5) = 75 + 150 = 225$

Since the total cost is $225 for both companies when *h* = 5, our answer is correct.

#### Key Concepts and Strategies

* **Linear Equations:** Recognizing and forming linear equations from word problems.
* **Systems of Equations:** Solving for a variable by setting two equations equal to each other.
* **Variable Definition:** Clearly defining your variables is crucial for setting up the equations correctly.
* **Verification:** Always verify your answer by plugging it back into the original equations to ensure it makes sense.

#### Common Mistakes to Avoid

* **Misinterpreting the Word Problem:** Read the problem carefully to correctly identify the flat fee and hourly rate for each company.
* **Incorrectly Setting Up Equations:** Double-check that you have the correct coefficients and constants in your equations.
* **Algebra Errors:** Be careful with your algebraic manipulations when solving for *h*.

#### Real-World Application

This type of problem is a practical example of how linear equations can be used to compare costs and make informed decisions in real-life situations, such as choosing between different service providers.

### Conclusion

By mastering the techniques of defining variables, setting up equations, and solving for unknowns, you’ll be well-prepared to tackle similar algebra problems on the SAT. Remember to practice regularly and review your work to avoid common mistakes. Good luck!