Expressions, Equations, and Relationships>Writing One-Variable, Two-Step Equations and Inequalities(TEKS.Math.7.10.A)
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Texas 7th Grade Math › Expressions, Equations, and Relationships>Writing One-Variable, Two-Step Equations and Inequalities(TEKS.Math.7.10.A)
Which real-world scenario matches the equation $3x + 7 = 22$?
A gym charges 7 dollars per visit plus a 3-dollar membership fee. You paid 22 dollars total. How many visits $x$ did you make?
A phone case costs 3 dollars, and sales tax is 7 dollars. The total is 22 dollars.
A ride costs 3 dollars per mile, plus a 7-dollar pickup fee. The total was 22 dollars. How many miles $x$ did you travel?
You had 3 dollars and then saved 7 dollars each day for $x$ days to reach 22 dollars.
Explanation
The equation $3x + 7 = 22$ means 3 dollars times the number of miles ($x$) plus a flat 7-dollar fee equals a total of 22 dollars. Choice C states exactly that structure: total cost = (3 per mile)$\times x$ + 7, and it equals 22.
A music app charges a \$10 monthly fee plus \$1.25 per song download. Jordan's budget allows at most \$35 this month. Let $s$ be the number of songs. What inequality represents this constraint?
$10 + 1.25s \ge 35$
$1.25 + 10s \le 35$
$10 + 1.25s \le 35$
$1.25s = 35$
Explanation
The fixed amount (constant) is $10$ and the rate (coefficient) is \$1.25$ per song, so the expression is $10 + 1.25s$. "At most \$35" means the total cannot exceed 35, so use $\le$: \$10 + 1.25s \le 35$.
Solve: $5x - 8 = 27$. Which value of $x$ makes this equation true?
5
6
7
8
Explanation
Add 8 to both sides to undo the subtraction: $5x = 35$. Then divide both sides by 5: $x = 7$. Check: $5(7) - 8 = 35 - 8 = 27$, which is true.
A triangular prism has base area 30 cm² and height 12 cm, giving volume 360 cm³. A triangular pyramid with the same base and the same height has volume 120 cm³.
What is the relationship between these volumes when the prism and pyramid share congruent bases and heights?
They are always equal if the base and height match.
The pyramid's volume is 1/3 of the prism's volume.
The pyramid's volume is 3 times the prism's volume.
The prism uses $V=\frac{1}{3}Bh$ while the pyramid uses $V=Bh$.
Explanation
For congruent bases and heights, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so the pyramid has $\tfrac{1}{3}$ the volume of the prism. Geometrically, three identical pyramids with the same base and height can fill the prism exactly.
The angles in a triangle measure $x^\circ$, $(2x + 15)^\circ$, and $(x - 5)^\circ$. Which equation can you write using the triangle angle-sum relationship?
$x + (2x + 15) = (x - 5)$
$x + (2x + 15) + (x - 5) = 180$
$x + (2x + 15) + (x - 5) = 90$
$(2x + 15) = (x - 5)$
Explanation
Triangle angle sum: $180^\circ$. Set up $x + (2x + 15) + (x - 5) = 180$. Simplify: $4x + 10 = 180 \Rightarrow 4x = 170 \Rightarrow x = 42.5$. The angles are $42.5^\circ$, $100^\circ$, and $37.5^\circ$, which add to $180^\circ$. Distractors either set angles equal or use $90^\circ$ incorrectly.
Solve the inequality: $3y + 12 \le 30$. What is the solution for $y$?
$y \le 6$
$y \ge 6$
$y \le -6$
$y < 6$
Explanation
Subtract 12 from both sides: $3y \le 18$. Divide both sides by 3: $y \le 6$. No inequality flip is needed because we divided by a positive number. Check with $y=6$: $3(6)+12=18+12=30$, which satisfies $\le 30$.
Two supplementary angles measure $(3y + 20)^\circ$ and $(2y - 10)^\circ$. What is the measure of each angle?
110° and 70°
95° and 85°
128° and 52°
122° and 58°
Explanation
Supplementary angles sum to $180^\circ$. Set up $(3y + 20) + (2y - 10) = 180$. Then $5y + 10 = 180 \Rightarrow 5y = 170 \Rightarrow y = 34$. Angles: $3(34)+20 = 122^\circ$ and $2(34)-10 = 58^\circ$. Check: $122^\circ + 58^\circ = 180^\circ$.
Which situation could be represented by the inequality $2y - 5 < 13$?
A craft store charges 2 dollars per yard of ribbon, and you use a 5-dollar coupon. You want to spend less than 13 dollars. How many yards $y$ can you buy?
A concert charges 2 dollars per ticket after a 5-dollar coupon, and you plan to spend at least 13 dollars. How many tickets $y$ can you buy?
A gym charges 5 dollars per visit plus a 2-dollar fee. You want the total to be under 13 dollars. How many visits $y$ can you make?
You save 2 dollars each day for $y$ days and then lose 5 dollars; your savings equal 13 dollars.
Explanation
The inequality $2y - 5 < 13$ translates to (2 dollars per yard)$\times y$ minus a 5-dollar coupon is less than 13 dollars. Choice A matches multiply by $y$, subtract 5, and use "less than 13." Other choices change the operations or the inequality type.
A ride service charges a \$3 base fare plus \$2.20 per mile. The total cost of Mateo's ride was $19.20. Let $m$ be the number of miles. Which equation models this situation?
$3 + 2.20m = 19.20$
$2.20 + 3m = 19.20$
$3 + 2.20m \le 19.20$
$2.20m = 19.20$
Explanation
The fixed amount (constant) is $3$ and the rate (coefficient) is \$2.20$ per mile, so the total cost is $3 + 2.20m$. Because the total was exactly \$19.20$, use $=$: $3 + 2.20m = 19.20$.
Consider the formulas for solids with the same triangular base area $B$ and height $h$: prism volume $V=Bh$ and triangular pyramid volume $V=\frac{1}{3}Bh$.
How do the formulas $V=Bh$ (prism) and $V=\frac{1}{3}Bh$ (pyramid) show the relationship between their volumes when bases and heights match?
They show the pyramid has 3 times the prism's volume because $\frac{1}{3}Bh$ is larger than $Bh$.
They show $B$ must be tripled for the pyramid to match the prism.
They show the volumes can't be compared without slant height.
They show the pyramid's volume is $\frac{1}{3}$ of the prism's volume for the same $B$ and $h$.
Explanation
With the same $B$ and $h$, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so $V_{\text{pyramid}}=\tfrac{1}{3}V_{\text{prism}}$. Geometrically, three such pyramids fit exactly into the prism.