Precalculus Practice: Evaluating, Simplifying, and Translating Algebraic Expressions

Q2. Evaluate 8 + 6x for x = 5

Background

Topic: Evaluating Algebraic Expressions

This question tests your ability to substitute a given value for a variable and perform arithmetic operations to evaluate the expression.

Key Terms and Formulas

• Algebraic Expression: An expression that contains numbers, variables, and operations.

• Substitution: Replacing a variable with a given value.

Step-by-Step Guidance

1. Identify the value of the variable: $x = 5$.

2. Substitute $x$ with 5 in the expression: $8 + 6x$ becomes $8 + 6 \times 5$.

3. Multiply $6 \times 5$ to find the product.

4. Add the result to 8 to complete the evaluation.

Try solving on your own before revealing the answer!

Final Answer: 38

$8 + 6 \times 5 = 8 + 30 = 38$

After substituting and performing the operations, the value of the expression is 38.

Q4. Evaluate 3x - y for x = 3 and y = 8

Background

Topic: Evaluating Algebraic Expressions with Two Variables

This question checks your ability to substitute values for two variables and perform the required arithmetic operations.

Key Terms and Formulas

• Substitution: Replace each variable with its given value.

Step-by-Step Guidance

1. Identify the values: $x = 3$, $y = 8$.

2. Substitute these values into the expression: $3x - y$ becomes $3 \times 3 - 8$.

3. Multiply $3 \times 3$ to get the product.

4. Subtract 8 from the result of the multiplication.

Try solving on your own before revealing the answer!

Final Answer: 1

$3 \times 3 - 8 = 9 - 8 = 1$

After substituting and simplifying, the value is 1.

Q18. Convert 86°F to Celsius using the formula $C = \frac{5}{9}(F - 32)$

Background

Topic: Temperature Conversion (Fahrenheit to Celsius)

This question tests your ability to use a linear formula to convert temperatures between Fahrenheit and Celsius scales.

Key Terms and Formulas

• Fahrenheit (F): A temperature scale.

• Celsius (C): Another temperature scale.

• Conversion Formula: $C = \frac{5}{9}(F - 32)$

Step-by-Step Guidance

1. Identify the given Fahrenheit temperature: $F = 86$.

2. Substitute $F$ into the formula: $C = \frac{5}{9}(86 - 32)$.

3. Calculate $86 - 32$ to find the difference.

4. Multiply the result by $\frac{5}{9}$ to find the Celsius temperature.

Try solving on your own before revealing the answer!

Final Answer: 30°C

$C = \frac{5}{9}(86 - 32) = \frac{5}{9}(54) = 30$

So, 86°F is equivalent to 30°C.

Q20. What was the ball’s height 3 seconds after it was kicked? (Use $h = 4 + 60t - 16t^2$)

Background

Topic: Evaluating Quadratic Functions

This question tests your ability to substitute a value into a quadratic function and perform the necessary calculations to find the output.

Key Terms and Formulas

• Quadratic Function: A function of the form $h = at^2 + bt + c$.

• Substitution: Replace $t$ with the given value.

Step-by-Step Guidance

1. Identify the value of $t$: $t = 3$.

2. Substitute $t$ into the formula: $h = 4 + 60t - 16t^2$ becomes $h = 4 + 60 \times 3 - 16 \times 3^2$.

3. Calculate $60 \times 3$ and $3^2$.

4. Multiply $16$by the result of $3^2$.

5. Add and subtract the results as indicated in the formula to find $h$.

Try solving on your own before revealing the answer!

Final Answer: -14 feet

$h = 4 + 60 \times 3 - 16 \times 3^2 = 4 + 180 - 16 \times 9 = 4 + 180 - 144 = 184 - 144 = 40$

Wait, let's check: $16 \times 9 = 144$, $4 + 180 = 184$, $184 - 144 = 40$. So the height is 40 feet.

Q86. Simplify $2(5x + 4) - 3$

Background

Topic: Simplifying Algebraic Expressions

This question tests your ability to use the distributive property and combine like terms.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Like Terms: Terms that have the same variable raised to the same power.

Step-by-Step Guidance

1. Apply the distributive property to $2(5x + 4)$.

2. Multiply $2$by $5x$ and $2$by $4$.

3. Write the expression after distribution and subtract 3.

4. Combine like terms if possible.

Try solving on your own before revealing the answer!

Final Answer: $10x + 5$

$2(5x + 4) - 3 = 10x + 8 - 3 = 10x + 5$

After distributing and combining like terms, the simplified expression is $10x + 5$.

Q112. Simplify $10^2 - 100 \div 5^2 - 2 \cdot 3$

Background

Topic: Order of Operations (PEMDAS)

This question tests your ability to correctly apply the order of operations to simplify an expression.

Key Terms and Formulas

• Order of Operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Step-by-Step Guidance

1. Evaluate exponents: $10^2$ and $5^2$.

2. Perform division: $100 \div$ (result from $5^2$).

3. Multiply $2 \cdot 3$.

4. Subtract the results in the correct order.

Try solving on your own before revealing the answer!

Final Answer: 75

$10^2 - 100 \div 5^2 - 2 \cdot 3 = 100 - 100 \div 25 - 6 = 100 - 4 - 6 = 90$

After following the order of operations, the simplified value is 90.

Q121. Write and simplify: A number decreased by the sum of the number and four

Background

Topic: Translating English Phrases to Algebraic Expressions

This question tests your ability to convert a verbal statement into an algebraic expression and then simplify it.

Key Terms and Formulas

• Let $x$ represent the number.

Step-by-Step Guidance

1. Translate "a number" as $x$.

2. Translate "the sum of the number and four" as $x + 4$.

3. "A number decreased by" means subtract the sum from the number: $x - (x + 4)$.

4. Simplify the expression by distributing the negative sign and combining like terms.

Try solving on your own before revealing the answer!

Final Answer: $-4$

$x - (x + 4) = x - x - 4 = -4$

After translating and simplifying, the result is $-4$.