Expressions, Equations, and Relationships>Modeling and Solving Equations with Variables on Both Sides Using Rational Numbers(TEKS.Math.8.8.C)
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Texas 8th Grade Math › Expressions, Equations, and Relationships>Modeling and Solving Equations with Variables on Both Sides Using Rational Numbers(TEKS.Math.8.8.C)
Solve $5x - 8 = 3x + 12$. What is the solution?
2
-10
10
20
Explanation
Subtract $3x$ from both sides: $2x - 8 = 12$. Add 8 to both sides: $2x = 20$. Divide by 2: $x = 10$. Check: Left side $= 5(10) - 8 = 50 - 8 = 42$; right side $= 3(10) + 12 = 30 + 12 = 42$. Both sides match, so $x=10$.
Two car rental plans: Plan A costs 15 dollars plus 0.35 dollars per mile. Plan B costs 5 dollars plus 0.45 dollars per mile. At how many miles are the costs equal?
100
10
50
105
Explanation
Let $m$ be miles. Set costs equal: $15 + 0.35m = 5 + 0.45m$. Subtract $0.35m$: $15 = 5 + 0.10m$. Subtract 5: $10 = 0.10m$. Divide by $0.10$: $m = 100$. Check: Plan A $= 15 + 0.35(100) = 15 + 35 = 50$; Plan B $= 5 + 0.45(100) = 5 + 45 = 50$. Equal at $100$ miles.
Solve $\frac{3}{4}x + 6 = \frac{1}{2}x + 12$. What is the solution?
12
6
-24
24
Explanation
Subtract $\tfrac{1}{2}x$ from both sides: $\left(\tfrac{3}{4} - \tfrac{1}{2}\right)x + 6 = 12 \Rightarrow \tfrac{1}{4}x + 6 = 12$. Subtract 6: $\tfrac{1}{4}x = 6$. Multiply by 4: $x = 24$. Check: Left $= \tfrac{3}{4}(24) + 6 = 18 + 6 = 24$; Right $= \tfrac{1}{2}(24) + 12 = 12 + 12 = 24$.
Solve $2(3x - 5) + 4 = x + 9$. What is the solution?
5
3
-3
15
Explanation
Distribute: $6x - 10 + 4 = x + 9 \Rightarrow 6x - 6 = x + 9$. Subtract $x$: $5x - 6 = 9$. Add 6: $5x = 15$. Divide by 5: $x = 3$. Check: Left $= 2(3\cdot 3 - 5) + 4 = 2(9 - 5) + 4 = 2\cdot 4 + 4 = 12$; Right $= 3 + 9 = 12$.
Two phone data plans: Plan M charges 25 dollars per month plus 2 dollars per GB of data. Plan N charges 10 dollars per month plus 3.5 dollars per GB. At how many gigabytes are the costs equal?
10
15
7.5
22.5
Explanation
Let $g$ be gigabytes. Set costs equal: $25 + 2g = 10 + 3.5g$. Subtract $2g$: $25 = 10 + 1.5g$. Subtract 10: $15 = 1.5g$. Divide by $1.5$: $g = 10$. Check: Plan M $= 25 + 2(10) = 45$; Plan N $= 10 + 3.5(10) = 45$.