Expressions, Equations, and Relationships
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Texas 6th Grade Math › Expressions, Equations, and Relationships
Situation: A dog-walking service charges a \$5 booking fee plus \$12 per hour. Which equation represents this situation, where x is the number of hours and y is the total cost?
Equation: $y = 12x + 5$
Table: x|y -> 0|12; 1|24; 2|36; 3|48
Graph description: a straight line passing through the origin with slope 12
Verbal: The total cost is a \$12 fee plus \$5 for each hour
Explanation
The \$5 is a fixed fee (y-intercept), and \$12 per hour is the rate (slope). So $y = 12x + 5$. The table and graph that start at $y=0$ ignore the fixed fee, and the swapped-fee verbal description is incorrect.
Notebooks $n$: 0, 2, 4, 6 Cost (dollars) $c$: 0, 3, 6, 9
What equation shows how $c$ depends on $n$?
$n=1.5c$
$c=1.5n$
$c=n+1.5$
$c=2n$
Explanation
Cost increases by 3 dollars every 2 notebooks, which is \$1.5$ per notebook, and $c=0$ when $n=0$. So $c=1.5n$.
Write a real-world problem that matches the equation $x + 7 = 15$.
Diego had some marbles. After buying 7 more, he had 15 marbles in all. How many marbles did he have to start with?
Diego had 15 marbles. He gave away 7. How many marbles are left?
Diego has 7 times as many marbles as his friend, and together they have 15 marbles. How many does his friend have?
When Diego adds 15 to a number, the result is 7. What is the number?
Explanation
The expression $x + 7$ means starting amount plus 7 more, and the equation equals 15 total. So the scenario of starting with $x$ marbles, buying 7 more, and ending with 15 matches $x + 7 = 15$. The other options use subtraction, multiplication, or reverse the numbers.
A parallelogram has a base of 7.2 meters and a height of 4.5 meters. Which equation shows the area of the parallelogram?
$A = 2(7.2 + 4.5)$
$A = 7.2 \times 4.5$
$A = \tfrac{1}{2} \times 7.2 \times 4.5$
$A = 7.2 \times 7.2$
Explanation
The area of a parallelogram is $A = b \times h$. Substituting: $A = 7.2 \times 4.5$.
Table: Hours worked and total pay Hours worked: 1 | 2 | 3 | 4 Total pay ($): 12 | 24 | 36 | 48
Which quantity is independent?
Total pay
Hours worked
Both are independent
Neither is independent
Explanation
Hours worked can be chosen freely, and the total pay changes in response. Therefore, hours worked is the independent quantity and total pay is the dependent quantity.
What is the value of this expression? $$3^2 + 5 \times (12 - 8)$$
$29$
$26$
$56$
$61$
Explanation
Use order of operations (PEMDAS). Parentheses: $12-8=4$. Exponents: $3^2=9$. Multiplication: $5\times4=20$. Addition: $9+20=29$. Distractors: $26$ treats $3^2$ as $6$; $56$ adds before multiplying: $(3^2+5)\times4$; $61$ ignores the parentheses: $5\times12-8=60-8=52$, then $52+9=61$.
Which statement correctly compares $3x+5$ and $3x+5=14$?
Both are equations because they both have numbers and variables.
Both are expressions because they both have operations.
$3x+5$ is an equation you can solve; $3x+5=14$ is just a phrase.
$3x+5$ is an expression (no equal sign), and $3x+5=14$ is an equation (has an equal sign).
Explanation
Expressions are mathematical phrases without equal or inequality signs. Equations are mathematical sentences that include an equal sign. So $3x+5$ is an expression and $3x+5=14$ is an equation.
What is the equivalent expression for $7a + 3a$?
$21a$
$10a$
$a^{10}$
$a$
Explanation
Factor the common $a$ using the distributive property: $7a + 3a = (7+3)a = 10a$. The distributive property justifies rewriting $ab+ac$ as $(b+c)a$ (factoring), then add $7+3=10$.
The perimeter of a square is 32 cm. What is the side length of the square?
8 cm
4 cm
32 cm
16 cm
Explanation
Let $x$ be the side length. The perimeter of a square is $4x$, so set up $4x = 32$. Divide both sides by 4 to get $x = 8$. Check: $4 \times 8 = 32$, which matches the given perimeter.
A parallelogram has a base of 7.5 meters and a height of 4.2 meters. What is the area?
31.5 m^2
15.75 m^2
11.7 m^2
31.5 m^3
Explanation
Use the area formula for a parallelogram: $A = b \times h$. Substitute: $A = 7.5 \times 4.2 = 31.5$. Attach units for area: $31.5\ \text{m}^2$.