Expressions, Equations, and Relationships
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Texas 8th Grade Math › Expressions, Equations, and Relationships
The system $y=2x+3$ and $y=-x+9$ is graphed. What is the intersection point?
(7, 2)
(0, 3)
(2, 7)
(6, 15)
Explanation
Set the equations equal: $2x+3=-x+9 \Rightarrow 3x=6 \Rightarrow x=2$. Then $y=2(2)+3=7$. Check: In $y=2x+3$, $y=7$; in $y=-x+9$, $y=-2+9=7$. So $(2,7)$ satisfies both. The intersection is the $(x,y)$ that makes both equations true.
A ladder leans against a wall. The ladder is 13 feet long, and the base is 5 feet from the wall. How high up the wall does the ladder reach?
12 feet
13 feet
5 feet
14 feet
Explanation
The ladder is the hypotenuse, and the wall height and ground distance are the legs. Use $a^2 + b^2 = c^2$ with $c = 13$ and a leg $a = 5$. Then $b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ feet. This models a common construction setup for checking heights safely.
Two cell plans charge a monthly fee plus a cost per gigabyte of data. Plan A charges a monthly fee of 15 dollars plus 4.50 dollars per GB. Plan B charges a monthly fee of 9 dollars plus 6 dollars per GB. Let $x$ be the number of gigabytes used in a month.
What equation shows when the two monthly costs are equal?
$15 + 4.5x = 9 + 6x$
$15 + 4.5 = 9 + 6x$
$4.5x = 9 + 6x$
$15 + 4.5x \le 9 + 6x$
Explanation
Equal costs means set Plan A's expression equal to Plan B's: $15 + 4.5x = 9 + 6x$. Choice B drops the variable on 4.5, C omits the monthly fee for Plan A, and D is an inequality instead of an equation.
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 $x=10$.
The system $y=3x-4$ and $y=3x+1$ is graphed. What is the intersection point?
No solution
(1, -1)
(-1, -7)
(0, 1)
Explanation
Both lines have the same slope $m=3$ but different $y$-intercepts ($-4$ and $1$), so they are parallel and never meet. No $(x,y)$ satisfies both at once. Therefore, there is no solution (no intersection).
Two gym membership options each have a one-time joining fee and a monthly cost. Plan 1 charges a 25 dollar joining fee plus 12 dollars per month. Plan 2 charges a 10 dollar joining fee plus 15 dollars per month. Let $x$ be the number of months.
Which equation represents the month count when the total costs are the same?
$25 + 12 = 10 + 15x$
$12x = 10 + 15x$
$25 + 12x = 10 + 15x$
$25 + 12x \ge 10 + 15x$
Explanation
Equal total cost means set Plan 1's total equal to Plan 2's: $25 + 12x = 10 + 15x$. A drops the variable on 12, B omits the joining fee for Plan 1, and D uses an inequality instead of an equation.
A rectangular field is 60 meters by 80 meters. What is the diagonal distance across the field?
140 meters
100 meters
60 meters
80 meters
Explanation
The diagonal is the hypotenuse of a right triangle with legs 60 and 80. $d = \sqrt{60^2 + 80^2} = \sqrt{3600 + 6400} = \sqrt{10000} = 100$ meters. This is useful in surveying and planning straight paths across rectangular lots.
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.
The system $y=\tfrac{1}{2}x-1$ and $y=-2x+5$ is graphed. What is the intersection point?
(2, 0)
(1/5, 12/5)
(8/5, -1/5)
(12/5, 1/5)
Explanation
Set equal: $\tfrac{1}{2}x-1=-2x+5 \Rightarrow \left(\tfrac{1}{2}+2\right)x=6 \Rightarrow \tfrac{5}{2}x=6 \Rightarrow x=\tfrac{12}{5}$. Then $y=\tfrac{1}{2}\cdot\tfrac{12}{5}-1=\tfrac{6}{5}-1=\tfrac{1}{5}$. Check in $y=-2x+5$: $-2\cdot\tfrac{12}{5}+5=-\tfrac{24}{5}+\tfrac{25}{5}=\tfrac{1}{5}$. The intersection is $(\tfrac{12}{5},\tfrac{1}{5})$. Intersection means the $(x,y)$ that satisfies both equations.
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 $= \tfrac{1}{2}(24) + 12 = 12 + 12 = 24$.