College Algebra Study Guidance: Solving Applied Problems and Triangle Perimeter Modeling

Q1. What are the steps for solving an applied problem in College Algebra?

Background

Topic: Problem-Solving Strategies in Algebra

This question is about understanding the systematic approach to solving real-world or applied problems using algebraic methods. These steps help organize your thinking and ensure you address all parts of the problem.

Key Terms:

• Variable: A symbol (often a letter) used to represent an unknown value.

• Equation: A mathematical statement that shows the equality between two expressions.

• Perimeter: The total distance around a shape.

Step-by-Step Guidance

1. Read the problem carefully to identify what information is given and what needs to be found.

2. Assign a variable to represent the unknown value. Clearly state what the variable stands for.

3. Write an equation using the variable(s) based on the relationships described in the problem.

4. Prepare to solve the equation for the unknown variable, but pause before completing the calculation.

Try solving on your own before revealing the answer!

Final Answer:

The steps for solving an applied problem are:

1. Read the problem carefully.

2. Assign a variable for the unknown.

3. Write an equation using the variable.

4. Solve the equation.

5. State the answer and check if it is reasonable.

6. Check the answer in the context of the original problem.

These steps help ensure you approach applied problems methodically and accurately.

Q2. Dimensions of a Puzzle Piece: A triangle-shaped puzzle piece has a perimeter of 30 cm. Two sides are each twice as long as the shortest side. Find the length of the shortest side.

Background

Topic: Modeling with Linear Equations

This question tests your ability to translate a geometric description into an algebraic equation and solve for an unknown side length using perimeter relationships.

Key Terms and Formulas:

• Perimeter of a triangle: $P = a + b + c$

• Let $x$ be the length of the shortest side.

• The other two sides are $2x$ and $2x$ (twice as long as the shortest side).

Step-by-Step Guidance

1. Assign a variable: Let $x$ represent the length of the shortest side (in cm).

2. Express the other sides in terms of $x$: The other two sides are $2x$ each.

3. Write the perimeter equation: $x + 2x + 2x = 30$

4. Simplify the equation to combine like terms, but pause before solving for $x$.

Try solving on your own before revealing the answer!

Final Answer: The shortest side is 6 cm.

Combine the terms: $x + 2x + 2x = 5x = 30$

Solve for $x$: $x = \frac{30}{5} = 6$

The shortest side of the triangle is 6 cm. This makes the other sides 12 cm each, and the total perimeter is $6 + 12 + 12 = 30$ cm, which matches the given perimeter.