Mixed Questions Set: Algebra Foundations

Ans: (b)
Explanation: A variable is a letter or symbol (such as \(x\), \(y\), or \(a\)) used to represent an unknown or changeable number. Option (a) describes a constant, option (c) describes a solution, and option (d) describes an operation.

Ans: (a)
Explanation: Combine like terms: the \(x\)-terms are \(3x + 2x = 5x\), and the constant terms are \(5 + (-3) = 2\). The simplified expression is \(5x + 2\). Option (b) incorrectly adds the constants; option (c) incorrectly combines the coefficients; option (d) ignores the constant terms.

Ans: (b)
Explanation: Use the distributive property: \(4(2x + 3) = 4 \cdot 2x + 4 \cdot 3 = 8x + 12\). Option (a) incorrectly adds instead of distributing; option (c) fails to distribute the 4 to the 3; option (d) incorrectly distributes.

Ans: (a)
Explanation: To solve \(x + 7 = 12\), subtract 7 from both sides: \(x = 12 - 7 = 5\). This uses the subtraction property of equality. Option (b) incorrectly adds; option (c) gives the opposite sign; option (d) results from a calculation error.

Ans: (a)
Explanation: To solve \(3x = 21\), divide both sides by 3: \(x = \frac{21}{3} = 7\). This uses the division property of equality. Option (b) subtracts incorrectly; option (c) adds; option (d) multiplies instead of dividing.

Ans: operations (or mathematical operations)
Explanation: An algebraic expression combines variables, constants (numbers), and operations such as addition, subtraction, multiplication, and division. Examples include \(2x + 5\) and \(3a - 7b\).

Ans: coefficient
Explanation: A coefficient is the number that multiplies a variable. In \(5x\), the coefficient is 5. It tells us how many groups of \(x\) we have.

Ans: variable
Explanation: Like terms have identical variable parts. For example, \(3x\) and \(7x\) are like terms because they both contain \(x\) to the first power. The terms \(2x\) and \(2y\) are not like terms because they have different variables.

Ans: isolate
Explanation: Inverse operations are operations that undo each other (like addition and subtraction, or multiplication and division). We use them to isolate the variable so we can find its value. For example, to solve \(x - 4 = 10\), we add 4 to both sides to isolate \(x\).

Ans: equation (first blank); solution (second blank)
Explanation: An equation is a mathematical statement showing that two expressions are equal. A solution is the value of the variable that makes the equation true. When \(x = 4\), we have \(2(4) + 3 = 8 + 3 = 11\), so the equation is true.

Ans:
Step 1: Identify the cost of each item. Tickets cost $12 each, and popcorn costs $8 each.
Step 2: Write the expression for \(t\) tickets and 2 popcorns.
Total cost expression: \(12t + 8(2) = 12t + 16\)
Step 3: Substitute \(t = 5\) into the expression.
\[12(5) + 16 = 60 + 16 = 76\]
Final Answer: The algebraic expression is \(12t + 16\) dollars. When you buy 5 tickets, the total cost is $76.

Ans:
Step 1: Identify the values. Length \(l = 15\) cm and width \(w = 8\) cm.
Step 2: Substitute into the perimeter formula.
\[P = 2(15) + 2(8)\]
Step 3: Perform the multiplications.
\[P = 30 + 16\]
Step 4: Add the results.
\[P = 46\]
Final Answer: The perimeter of the rectangle is 46 cm.

Ans:
Step 1: Identify and group like terms.
Variable terms: \(6a - 2a + 3a\)
Constant terms: \(8 + 5\)
Step 2: Combine like terms.
\[6a - 2a + 3a = 7a\]
\[8 + 5 = 13\]
Step 3: Write the simplified expression.
\[7a + 13\]
Step 4: Substitute \(a = 2\).
\[7(2) + 13 = 14 + 13 = 27\]
Final Answer: The simplified expression is \(7a + 13\). When \(a = 2\), the value is 27.

Ans:
Step 1: Subtract 5 from both sides to isolate the term with the variable.
\[2x + 5 - 5 = 19 - 5\]
\[2x = 14\]
Step 2: Divide both sides by 2.
\[x = \frac{14}{2} = 7\]
Step 3: Check the solution by substituting \(x = 7\) back into the original equation.
\[2(7) + 5 = 14 + 5 = 19\] ✓
The solution is correct.
Final Answer: \(x = 7\)

Ans:
Step 1: Translate the words into an algebraic expression. "3 more pencils than notebooks" means the number of pencils equals the number of notebooks plus 3.
Expression for pencils: \(n + 3\)
Step 2: Substitute \(n = 7\).
\[7 + 3 = 10\]
Final Answer: The expression is \(n + 3\) pencils. When the student has 7 notebooks, the student has 10 pencils.

Ans:
Step 1: Apply the distributive property to the first group.
\[3(x + 4) = 3x + 12\]
Step 2: Apply the distributive property to the second group.
\[2(x - 1) = 2x - 2\]
Step 3: Combine the expanded expressions.
\[3x + 12 + 2x - 2\]
Step 4: Combine like terms.
\[3x + 2x = 5x\]
\[12 + (-2) = 10\]
Final Answer: The simplified expression is \(5x + 10\).

Ans:
Claim: Carlos solved the equation correctly. The solution to \(4x - 6 = 10\) is \(x = 4\), not \(x = 3\).
Evidence: When we solve \(4x - 6 = 10\) using the properties of equality, we first add 6 to both sides: \(4x - 6 + 6 = 10 + 6\), giving us \(4x = 16\). Then we divide both sides by 4: \(x = \frac{16}{4} = 4\). To verify, we check by substituting into the original equation: \(4(4) - 6 = 16 - 6 = 10\) ✓. If we check Carlos's answer of \(x = 3\): \(4(3) - 6 = 12 - 6 = 6 \neq 10\), so it does not satisfy the equation.
Reasoning: The solution to an equation is the value that makes the equation true when substituted back. Carlos's answer satisfies the original equation, making it correct. Maya's answer, \(x = 3\), produces a false statement (\(6 = 10\)), which means Maya likely made an error in isolating \(x\)-possibly by forgetting to divide by 4 or miscalculating after adding 6.

Ans:
Claim: The claim is incorrect. The expressions \(2(x + 3)\) and \(2x + 3\) are not equivalent.
Evidence: Using the distributive property, we expand \(2(x + 3) = 2 \cdot x + 2 \cdot 3 = 2x + 6\). This is different from \(2x + 3\). We can verify by testing a specific value: if \(x = 1\), then \(2(1 + 3) = 2(4) = 8\), while \(2(1) + 3 = 2 + 3 = 5\). Since \(8 \neq 5\), the expressions produce different results.
Reasoning: It is critical to apply the distributive property correctly when expanding expressions. The 2 must multiply both the \(x\) and the 3, not just the \(x\). When solving equations or evaluating expressions, using the wrong form leads to incorrect solutions. Simplifying correctly ensures that we work with equivalent expressions that represent the same mathematical relationships.

Ans:
Part A: Algebraic Solution
Step 1: Multiply both sides by 2 to eliminate the fraction.
\[2 \cdot \frac{x + 5}{2} = 2 \cdot 9\]
\[x + 5 = 18\]
Step 2: Subtract 5 from both sides to isolate \(x\).
\[x + 5 - 5 = 18 - 5\]
\[x = 13\]
Step 3: Check the solution by substituting \(x = 13\) into the original equation.
\[\frac{13 + 5}{2} = \frac{18}{2} = 9\] ✓

Part B: Real-World Scenario and Interpretation
Scenario: A group of friends collected money for charity. After combining their contributions and adding $5 from a fundraiser, they had a total of $18 to donate. The average amount each person would receive credit for is $9. How many friends contributed to the collection?
In equation form: The sum of contributions plus $5, divided by the number of contributors, equals $9.
Interpretation: The solution \(x = 13\) means that the original collected amount before adding the fundraiser contribution was $13. If we say 13 is also related to another context (like cents per person or the actual dollar amount before the $5 boost), then \(x = 13\) represents a meaningful quantity in the real-world scenario. The interpretation depends on how we define \(x\) in our scenario, but the algebraic solution remains \(x = 13\).
Final Answer: \(x = 13\). In a real-world context, this could represent the original amount collected before additional contributions or fundraiser amounts were added.

Ans:
Step 1: Define variables and set up relationships.
Let \(w\) = width of the rectangle (in cm)
Then \(l = w + 3\) (length is 3 cm more than width)
Step 2: Use the perimeter formula and substitute the relationships.
\[P = 2l + 2w\]
\[54 = 2(w + 3) + 2w\]
Step 3: Apply the distributive property.
\[54 = 2w + 6 + 2w\]
Step 4: Combine like terms.
\[54 = 4w + 6\]
Step 5: Subtract 6 from both sides.
\[54 - 6 = 4w\]
\[48 = 4w\]
Step 6: Divide both sides by 4 to solve for \(w\).
\[w = \frac{48}{4} = 12\]
Step 7: Find the length using the relationship \(l = w + 3\).
\[l = 12 + 3 = 15\]
Step 8: Verify the answer by checking the perimeter.
\[P = 2(15) + 2(12) = 30 + 24 = 54\] ✓
Final Answer: The width of the rectangle is 12 cm and the length is 15 cm.