Unit 1 Algebra Basics – Homework 5: Evaluating Expressions (Answer Key)
Evaluating algebraic expressions is one of the first skills students master in Unit 1 Algebra Basics. But homework 5 focuses on applying the order of operations, substituting values for variables, and simplifying results correctly. This answer key not only provides the final answers but also walks through each step so that learners can see why each answer is correct and how to avoid common mistakes.
Introduction – Why Evaluating Expressions Matters
When you evaluate an expression, you replace the variables with given numbers and compute the result using the proper order of operations (PEMDAS/BODMAS). Mastering this process builds a solid foundation for:
- Solving equations later in the course.
- Understanding functions and graphing.
- Tackling real‑world problems that involve formulas (e.g., area, speed, cost).
Homework 5 is designed to reinforce these concepts through a variety of expressions, including those with parentheses, exponents, and negative numbers.
Step‑by‑Step Guide to Solving the Problems
Below is a systematic approach you can use for every problem in this homework set And that's really what it comes down to..
- Read the problem carefully – Identify the expression and the values assigned to each variable.
- Write down the substitution – Replace each variable with its numerical value, keeping the original symbols (e.g., “+”, “–”, “·”, “/”).
- Apply the order of operations:
- Parentheses/Brackets – Simplify inner groups first.
- Exponents/Orders – Compute powers and square roots.
- Multiplication and Division – Perform from left to right.
- Addition and Subtraction – Perform from left to right.
- Check for sign errors – Remember that subtracting a negative is the same as adding a positive.
- Write the final numeric answer – Verify by plugging the result back into the original expression (optional but helpful).
Following this checklist reduces careless errors and ensures consistency across all problems.
Homework 5 Problems and Detailed Answers
Problem 1
Expression: (3x + 7)
Given: (x = 4)
Solution:
- Substitute: (3(4) + 7)
- Multiply: (12 + 7)
- Add: 19
Answer: 19
Problem 2
Expression: (5y - 2y^{2})
Given: (y = -3)
Solution:
- Substitute: (5(-3) - 2(-3)^{2})
- Compute the exponent: ((-3)^{2}=9) → (5(-3) - 2(9))
- Multiply: (-15 - 18)
- Add (actually subtract): (-33)
Answer: (-33)
Problem 3
Expression: (\dfrac{2a^{2} - 4a}{a})
Given: (a = 5)
Solution:
- Substitute: (\dfrac{2(5)^{2} - 4(5)}{5})
- Square the 5: (5^{2}=25) → (\dfrac{2(25) - 20}{5})
- Multiply: (\dfrac{50 - 20}{5})
- Subtract inside numerator: (\dfrac{30}{5})
- Divide: 6
Answer: 6
Problem 4
Expression: (4(m - 2) + 3m^{2})
Given: (m = -1)
Solution:
- Substitute: (4((-1) - 2) + 3(-1)^{2})
- Inside parentheses: ((-1) - 2 = -3) → (4(-3) + 3(1))
- Multiply: (-12 + 3)
- Add: -9
Answer: (-9)
Problem 5
Expression: ((p + 6)(p - 2))
Given: (p = 3)
Solution:
- Substitute: ((3 + 6)(3 - 2))
- Simplify each parenthesis: ((9)(1))
- Multiply: 9
Answer: 9
Problem 6
Expression: (\dfrac{7 - 2q}{q^{2}})
Given: (q = -2)
Solution:
- Substitute: (\dfrac{7 - 2(-2)}{(-2)^{2}})
- Multiply inside numerator: (2(-2) = -4) → (7 - (-4) = 7 + 4)
- Numerator becomes 11; denominator: ((-2)^{2}=4)
- Fraction: (\dfrac{11}{4}) → 2.75 (or ( \frac{11}{4}) as an exact fraction)
Answer: (\frac{11}{4}) or 2.75
Problem 7
Expression: (-3r^{3} + 4r^{2} - r)
Given: (r = 2)
Solution:
- Substitute: (-3(2)^{3} + 4(2)^{2} - 2)
- Compute powers: (2^{3}=8), (2^{2}=4) → (-3(8) + 4(4) - 2)
- Multiply: (-24 + 16 - 2)
- Combine: (-24 + 16 = -8); (-8 - 2 = -10)
Answer: (-10)
Problem 8
Expression: (\dfrac{(2s + 5)^{2}}{s})
Given: (s = -1)
Solution:
- Substitute: (\dfrac{(2(-1) + 5)^{2}}{-1})
- Inside parentheses: (2(-1) = -2); (-2 + 5 = 3) → (\dfrac{3^{2}}{-1})
- Square: (9) → (\dfrac{9}{-1})
- Divide: -9
Answer: (-9)
Problem 9
Expression: (8 - \dfrac{4t}{t - 1})
Given: (t = 3)
Solution:
- Substitute: (8 - \dfrac{4(3)}{3 - 1})
- Multiply numerator: (4(3)=12) → (8 - \dfrac{12}{2})
- Divide: (\dfrac{12}{2}=6)
- Subtract: (8 - 6 = 2)
Answer: 2
Problem 10
Expression: (\dfrac{(x + y)^{2} - (x - y)^{2}}{4xy})
Given: (x = 5,; y = 2)
Solution:
- Substitute: (\dfrac{(5 + 2)^{2} - (5 - 2)^{2}}{4(5)(2)})
- Compute inside parentheses: (5+2=7,; 5-2=3) → (\dfrac{7^{2} - 3^{2}}{40})
- Squares: (7^{2}=49,; 3^{2}=9) → (\dfrac{49 - 9}{40})
- Subtract numerator: (40) → (\dfrac{40}{40}=1)
Answer: 1
Common Pitfalls and How to Avoid Them
| Pitfall | Why It Happens | Fix |
|---|---|---|
Forgetting parentheses when substituting values (e.g., writing 3x+7 as 3*4+7 vs 3(4)+7). Consider this: |
Skipping the implicit multiplication sign. Consider this: | Always write the substitution with parentheses around the number: 3(4)+7. In practice, |
Misreading negative exponents (e. g., -2y^2 interpreted as (-2y)^2). |
Ambiguity in handwritten notes. On the flip side, | Follow the order: exponent applies only to the variable unless parentheses indicate otherwise. |
| Dividing by zero in expressions like (\frac{7-2q}{q^{2}}) when (q=0). | Not checking the denominator. Which means | Verify that the given value does not make any denominator zero; if it does, the expression is undefined. Practically speaking, |
| Incorrect sign handling when subtracting a negative number. | Tendency to treat “– (– a)” as “– a”. | Remember that “– (– a) = +a”. Write it explicitly if needed: -(-3) becomes +3. |
| Rounding too early (e.g.On the flip side, , converting (\frac{11}{4}) to 2. That said, 75 before completing other steps). | Desire for a decimal answer. | Keep fractions exact until the final step unless the problem explicitly asks for a decimal. |
Frequently Asked Questions (FAQ)
Q1: Do I need a calculator for these problems?
A: Not necessarily. All numbers in Homework 5 are small enough to compute by hand, which helps you practice mental arithmetic and order‑of‑operations skills. Use a calculator only to check your work.
Q2: What if the variable value is a fraction?
A: Substitute the fraction exactly (e.g., (x = \frac{3}{2})). Perform operations using fraction arithmetic to avoid rounding errors, then simplify the final result.
Q3: How can I verify my answers quickly?
A: After obtaining the final numeric answer, plug it back into the original expression (reverse the substitution) to see if both sides match. For more complex expressions, a quick calculator check can confirm your manual work.
Q4: Why does Problem 10 simplify to 1 regardless of the numbers?
A: The expression (\frac{(x+y)^{2}-(x-y)^{2}}{4xy}) is an algebraic identity that always equals 1 for any non‑zero (x) and (y). This is a good illustration of how recognizing patterns can save time.
Q5: What should I do if I get a negative denominator?
A: A negative denominator is perfectly valid; you can move the minus sign to the numerator or keep it in the denominator. The final value will be the same (e.g., (\frac{5}{-2} = -\frac{5}{2})) And it works..
Conclusion – Turning Practice into Mastery
Evaluating expressions is more than a routine homework task; it is the gateway to algebraic reasoning. By consistently applying the substitution steps, respecting the order of operations, and double‑checking for sign errors, you will:
- Gain confidence in handling increasingly complex formulas.
- Reduce the likelihood of careless mistakes on quizzes and tests.
- Build the intuition needed for solving equations, graphing functions, and tackling real‑world math problems.
Use this answer key as a learning tool, not just a list of results. Replicate the step‑by‑step process on new problems, and soon evaluating any algebraic expression will feel as natural as reading a sentence. Keep practicing, and the algebra basics covered in Unit 1 will become a solid foundation for all future math courses.
