Evaluate Expressions With Variables - 6th Grade Math

Evaluate the expression $\dfrac{x + 6}{2}$ when $x = 8$ .
Answer: 7 This question tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if $x=8$ in expression $(x+6)/2$ , write $(8+6)/2$ replacing x with 8, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: $(x+6)/2$ at $x=8$ becomes $(8+6)/2$ , parentheses: 14, divide: $14/2=7$ . With exponent: $x^2 - 5$ at $x=4$ = $16-5=11$ . Two variables: $2x+3y$ at $x=5$ , $y=2$ = $10+6=16$ . Specific evaluation: $(x+6)/2$ at $x=8$ , substitute: $(8+6)/2$ , evaluate: $14/2=7$ .
Evaluate $2x + 3y$ when $x = 5$ and $y = 2$ .
Answer: $16$ This problem tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=5 and y=2 in expression 2x+3y, write 2(5)+3(2) replacing x with 5 and y with 2, use parentheses for clarity). To evaluate 2x+3y when x=5 and y=2, substitute: 2(5)+3(2), then evaluate: 2×5=10 and 3×2=6 (multiply first), then 10+6=16 (add). The correct answer is 16. Common errors include not substituting all variables (leaving answer as 10+3y), concatenation (reading 2x as 25), or arithmetic mistakes. Process: (1) write expression with variables (2x+3y), (2) substitute all given values (x=5, y=2: write 2(5)+3(2)), (3) evaluate using order (multiply: 2×5=10 and 3×2=6, add: 10+6=16), (4) verify reasonable (2×5=10, 3×2=6, sum=16✓).
Marcus is saving money for a gaming console. He starts with $45 and saves $12 each week. After some number of weeks, he spends $30 on a game controller. The expression $45 + 12w - 30$ represents how much money Marcus has after $w$ weeks. What is the value of this expression when $w = 7$ ?
Answer: $99 Substitute $w = 7$ into the expression: $45 + 12(7) - 30 = 45 + 84 - 30 = 129 - 30 = 99$ .
Evaluate $x^2-5$ when $x=4$ .
Answer: 11 This problem is all about substitution—that means we swap the letter $x$ for the number it equals, then do the math. Here's how it works, one step at a time: First, write the expression: $x^2 - 5$ . Next, replace $x$ with $4$ : $4^2 - 5$ Now handle the exponent. Remember, $4^2$ means $4 \times 4$ , which is $16$ : $16 - 5$ Finally, subtract: $16 - 5 = 11$ So the answer is D) 11. Think of $x$ like a labeled box. The label says " $x = 4$ ," so whenever you see $x$ , you reach in and pull out a $4$ . Easy swap! Try this at home: Pick your own number for $x$ , like $x = 3$ , and evaluate $x^2 - 5$ again. Do the exponent first, then subtract. You've got this! 🎯
Find the value of $5 + 3(x - 2)^2$ when $x = 4$ .
Answer: 17 When you encounter an expression with variables and need to find its value for a specific number, you're working with substitution and order of operations. The key is to replace the variable with the given value and then carefully follow the order of operations (PEMDAS). Let's substitute $x = 4$ into the expression $5 + 3(x - 2)^2$ : $5 + 3(4 - 2)^2$ Now follow PEMDAS step by step. First, handle the parentheses: $4 - 2 = 2$ , giving us $5 + 3(2)^2$ . Next, calculate the exponent: $(2)^2 = 4$ , so we have $5 + 3(4)$ . Then multiply: $3 \times 4 = 12$ , leaving us with $5 + 12$ . Finally, add: $5 + 12 = 17$ . The answer is C. Let's see where the wrong answers come from. Choice A (41) likely results from incorrectly calculating $(4-2)^2$ as $4^2 - 2^2 = 16 - 4 = 12$ , then doing $5 + 3(12) = 41$ . Choice B (29) probably comes from substituting first to get $5 + 3(4-2)^2$ , but then mistakenly calculating $3 \times 4 - 2 = 12 - 2 = 10$ , squared to get $100$ , then somehow getting 29. Choice D (21) might result from forgetting to square the 2, calculating $5 + 3(4-2) = 5 + 3(2) = 5 + 6 = 11$ , or other order of operations errors. Always substitute first, then follow PEMDAS carefully. Write out each step to avoid rushing through the order of operations.
Evaluate $\frac{2x + y}{3} - x$ when $x = 6$ and $y = 9$ .
Answer: $1$ Substitute the values: $\frac{2(6) + 9}{3} - 6 = \frac{12 + 9}{3} - 6 = \frac{21}{3} - 6 = 7 - 6 = 1$ .
The cost in dollars to rent a moving truck is given by $25 + 0.50m$ , where $m$ is the number of miles driven. If someone drives 84 miles, what is the total cost?
Answer: $67 When you see an expression with variables like $25 + 0.50m$ , you're working with substitution – replacing the variable with a given number and calculating the result. This type of problem tests whether you can follow the order of operations while substituting values. To find the total cost when someone drives 84 miles, substitute $m = 84$ into the expression $25 + 0.50m$ : $25 + 0.50(84)$ First, multiply: $0.50 \times 84 = 42$ Then add: $25 + 42 = 67$ The total cost is $67, making B correct. Let's examine why the other answers are wrong. Choice A ($59) likely comes from incorrectly calculating $0.50 \times 84$ as $34 instead of $42, then adding $25 + 34 = 59$ . Choice C ( $42) is the result of only calculating the mileage portion ($ $0.50 \times 84 = 42$ $) but forgetting to add the base fee of $25. Choice D ($109) probably results from adding $25 + 84 = 109$ , completely ignoring the 0.50 multiplier – this shows misunderstanding of how the expression works. When substituting into expressions, always follow order of operations: handle multiplication before addition, and make sure you substitute the variable everywhere it appears. Double-check that you're using all parts of the original expression in your final calculation.
A movie theater charges $c = 4n + 6$ dollars for $n$ tickets (including a $6 fee). How much does it cost when $n = 3$ ?
Answer: $18$ This problem tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if n=3 in expression c=4n+6, write c=4(3)+6 replacing n with 3, use parentheses for clarity). To evaluate c=4n+6 when n=3, substitute: c=4(3)+6, then evaluate: 4×3=12 (multiply first), then 12+6=18 (add). The correct answer is $18. Common errors include order violations (adding before multiplying: 4×3+6 as 4×9=36), concatenation (reading 4n as 43), or arithmetic mistakes (12+6=17). Real-world: cost formula 4n+6 at n=3 tickets gives 4(3)+6=18 dollars (substitute n, evaluate, interpret as cost including $6 fee).
A rectangular garden has length $(3w + 4)$ feet and width $(2w - 1)$ feet. The expression for its perimeter is $2(3w + 4) + 2(2w - 1)$ . What is the perimeter when $w = 7$ ?
Answer: 76 feet Substitute $w = 7$ : $2(3(7) + 4) + 2(2(7) - 1) = 2(21 + 4) + 2(14 - 1) = 2(25) + 2(13) = 50 + 26 = 76$ feet.
Evaluate the expression $3x + 5$ when $x = 4$ .
Answer: 17 This question tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=4 in expression 3x+5, write 3(4)+5 replacing x with 4, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: 3x+5 at x=4 becomes 3(4)+5, multiply first: 12, then add: 12+5=17. With exponent: x²-5 at x=4 becomes 4²-5, exponent first: 16, subtract: 16-5=11. Two variables: 2x+3y at x=5, y=2 becomes 2(5)+3(2)=10+6=16. Specific evaluation: 3x+5 at x=4, substitute: 3(4)+5, evaluate: 3×4=12 (multiply), 12+5=17 (add).
A rectangle's area is $A = lw$ . If the length is $l = 9$ cm and the width is $w = 4$ cm, what is the area?
Answer: $36\text{ cm}^2$ Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if l=9, w=4 in expression lw, write 9×4 replacing them, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: lw at l=9, w=4 becomes 9×4=36. Specific evaluation: lw at l=9, w=4: substitute 9×4, evaluate 36 cm². Correct substitution and evaluation leads to 36 cm², which matches choice B. Common errors include adding instead of multiplying (9+4=13 as in A), or wrong operations like 9×4=36 but then extra (maybe $9^2$ ×4=324 as in D).
The expression $4n^2 - 3n + 7$ gives the number of tiles needed for a mosaic pattern when $n$ represents the row number. How many tiles are needed for row 5?
Answer: 92 tiles Substitute $n = 5$ : $4(5)^2 - 3(5) + 7 = 4(25) - 15 + 7 = 100 - 15 + 7 = 92$ tiles.
Verify: Is $3x + 2 = 20$ when $x = 6$ ?
Answer: Yes, because $3(6) + 2 = 20$ . This problem tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=6 in expression $3x+2$ , write $3(6)+2$ replacing x with 6, use parentheses for clarity). To verify if $3x+2=20$ when x=6, substitute: $3(6)+2$ , then evaluate: $3 \times 6 = 18$ (multiply first), then $18 + 2 = 20$ (add). Since $3(6)+2=20$ , the equation is true when x=6, so the answer is "Yes, because $3(6)+2=20$ ". Common errors include concatenation (reading 3x as 36, giving $36+2=38$ ), wrong order (adding before multiplying), or arithmetic mistakes ( $18+2=19$ ). Process: (1) substitute x=6 into $3x+2$ , (2) evaluate $3(6)+2=18+2=20$ , (3) compare to given value 20, (4) they match, so verification is correct.
A rectangle has length $x+2$ inches and width $3$ inches. The area is $A=3(x+2)$ . What is the area when $x=6$ ?
Answer: 24 Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if $x=6$ in $A=3(x+2)$ , write $3((6)+2)$ , use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: $3(x+2)$ at $x=6$ becomes $3(6+2)$ , parentheses: $8$ , multiply: $3 \times 8=24$ . For this specific question, evaluate $A=3(x+2)$ at $x=6$ : substitute $3(6+2)$ , parentheses: $6+2=8$ , multiply: $3 \times 8=24$ . Correct substitution and evaluation yield 24, which matches choice A. Real-world: rectangle area $3(x+2)$ at $x=6$ gives $3(8)=24$ square inches (substitute x, evaluate, interpret as area). Process: (1) write expression $3(x+2)$ , (2) substitute $x=6$ to get $3(6+2)$ , (3) evaluate using order (parentheses then multiply to get 24), (4) verify reasonable ( $6+2=8$ , times $3=24$ makes sense).
Evaluate the expression $3x+5$ when $x=4$ .
Answer: 17 Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=4 in expression 3x+5, write 3(4)+5 replacing x with 4, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: 3x+5 at x=4 becomes 3(4)+5, multiply first: 12, then add: 12+5=17. For this specific question, evaluate 3x+5 at x=4: substitute 3(4)+5, evaluate 3×4=12 (multiply), then 12+5=17 (add). Correct substitution and evaluation yield 17, which matches choice C. Common errors include concatenation (3x with x=4 as 34 instead of 3×4=12) or violating order by adding before multiplying (3+5=8, then 8×4=32). Process: (1) write expression 3x+5, (2) substitute x=4 to get 3(4)+5, (3) evaluate using order (multiply then add to get 17), (4) verify reasonable (3×4=12, +5=17 makes sense).
Substitute $x=5$ and $y=2$ into the expression $2x+3y$ and evaluate.
Answer: 16 Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (for x=5 and y=2 in 2x+3y, write 2(5)+3(2), use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Two variables: 2x+3y at x=5, y=2 becomes 2(5)+3(2)=10+6=16. For this specific question, substitute x=5 and y=2 into 2x+3y: 2(5)+3(2), evaluate 2×5=10 and 3×2=6 (multiply), then 10+6=16 (add). Correct substitution and evaluation yield 16, which matches choice A. Common errors include not substituting all variables (leaving as 10+3y) or arithmetic mistakes (10+6=15). Process: (1) write expression 2x+3y, (2) substitute values to get 2(5)+3(2), (3) evaluate using order (multiplies then add to get 16), (4) verify reasonable (2×5=10, 3×2=6, total 16 makes sense).
Substitute $x = 6$ into the expression $2x^2 - 3$ and evaluate.
Answer: $69$ This problem tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if x=6 in expression 2x²-3, write 2(6)²-3 replacing x with 6, use parentheses for clarity). To evaluate 2x²-3 when x=6, substitute: 2(6)²-3, then evaluate following PEMDAS: first exponent 6²=36, then multiply 2×36=72, finally subtract 72-3=69. The correct answer is 69. Common errors include treating x² as 2×x (so 6²=2×6=12 instead of 36), wrong order (subtracting before multiplying), or arithmetic mistakes. Process: (1) write expression with variables (2x²-3), (2) substitute given values (x=6: write 2(6)²-3), (3) evaluate using order (exponent: 6²=36, multiply: 2×36=72, subtract: 72-3=69), (4) verify reasonable (2×36≈72, minus 3≈69✓).
A store's daily profit is modeled by $150 + 25h - 5h^2$ , where $h$ is the number of hours past noon. What is the profit at 2:00 PM (when $h = 2$ )?
Answer: $180 When you encounter a problem involving a mathematical model or formula, your job is to substitute the given values and carefully perform the calculations step by step. Here, you need to find the profit when $h = 2$ (since 2:00 PM is 2 hours past noon). Substitute $h = 2$ into the profit formula $150 + 25h - 5h^2$ : $150 + 25(2) - 5(2)^2$ Work through this systematically: $25(2) = 50$ and $5(2)^2 = 5(4) = 20$ . So you have: $150 + 50 - 20 = 180$ The profit at 2:00 PM is $180, which is answer choice C. Let's see where the wrong answers might come from. Choice A ($190) likely results from forgetting to subtract the $5h^2$ term entirely, giving you $150 + 50 = 200$ , or making an arithmetic error. Choice B ($170) might occur if you miscalculate $5(2)^2$ as 10 instead of 20, giving you $150 + 50 - 10 = 190$ , or make another computational mistake. Choice D ($160) could result from incorrectly calculating $25(2)$ as 30 instead of 50, leading to $150 + 30 - 20 = 160$ . Remember: when substituting into algebraic expressions, follow the order of operations carefully. Calculate exponents first, then multiplication, and finally addition and subtraction from left to right. Double-check each arithmetic step to avoid careless errors.
A school club sells notebooks. The cost (in dollars) is given by $C=5n+3$ , where $n$ is the number of notebooks. What is the cost when $n=7$ ?
Answer: 38 Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (if n=7 in $C=5n+3$ , write $5(7)+3$ replacing n with 7, use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). Example: $5n+3$ at n=7 becomes $5(7)+3$ , multiply: $35$ , add: $35+3=38$ . For this specific question, evaluate $C=5n+3$ at n=7: substitute $5(7)+3$ , evaluate $5×7=35$ (multiply), then $35+3=38$ (add). Correct substitution and evaluation yield 38, which matches choice A. Real-world: cost formula $5n+3$ at n=7 items gives $5(7)+3=38$ dollars (substitute n, evaluate, interpret as cost). Process: (1) write expression $5n+3$ , (2) substitute n=7 to get $5(7)+3$ , (3) evaluate using order (multiply then add to get 38), (4) verify reasonable ( $5×7=35$ , +3=38 makes sense).
Substitute $x=3$ and $y=4$ into the expression $x^2+y$ and evaluate.
Answer: 13 Tests evaluating expressions by substituting variable values and applying order of operations (PEMDAS). Substitution: replace variables with given values (for x=3 and y=4 in x²+y, write (3)²+(4), use parentheses for clarity). Order of operations: evaluate following PEMDAS (parentheses first, exponents, multiply/divide, add/subtract). With exponent: x²+y at x=3, y=4 becomes 3²+4, exponent: 9, add: 9+4=13. For this specific question, substitute x=3 and y=4 into x²+y: 3²+4, evaluate exponent 3²=9, then 9+4=13 (add). Correct substitution and evaluation yield 13, which matches choice A. Common errors include exponent as multiplication (3×2=6 +4=10) or not substituting both variables. Process: (1) write expression x²+y, (2) substitute to get 3²+4, (3) evaluate using order (exponent then add to get 13), (4) verify reasonable (3 squared=9, +4=13 makes sense).

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