Proportionality>Representing Linear Proportional Situations with Tables, Graphs, and Equations(TEKS.Math.8.5.A)
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Texas 8th Grade Math › Proportionality>Representing Linear Proportional Situations with Tables, Graphs, and Equations(TEKS.Math.8.5.A)
A cyclist rides at a constant speed. The relationship between time $x$ (hours) and distance $y$ (miles) is shown by the points $(1, 60)$, $(2, 120)$, and $(3, 180)$. Which equation represents this proportional relationship?
y = 180x
y = 60x + 10
x = 60y
y = 60x
Explanation
For a proportional relationship, $y = kx$ and $k = \frac{y}{x}$. Using any point: $k = \frac{60}{1} = 60$, $\frac{120}{2} = 60$, $\frac{180}{3} = 60$. The constant ratio $\frac{y}{x}$ is 60, so $y = 60x$. Adding a constant (like $+10$) would make it non-proportional because it would not pass through the origin.
The cost of gasoline is proportional to the gallons purchased. If 5 gallons cost 17.50 and 8 gallons cost 28.00, which equation represents the relationship where $y$ is the total cost (dollars) and $x$ is the number of gallons?
y = 3.5x
y = 3.5x + 1
x = 3.5y
y = 0.35x
Explanation
In a proportional relationship, $y = kx$. Compute $k$ using $k = \frac{y}{x}$: $k = \frac{17.50}{5} = 3.5$ and $\frac{28.00}{8} = 3.5$. Since the ratio $\frac{y}{x}$ is constant, the equation is $y = 3.5x$. Adding a constant (like $+1$) would make it non-proportional.
A proportional relationship is graphed and passes through the origin and the point $(4, 10)$. Which equation represents this relationship?
y = 4x + 10
x = 2.5y
y = 2.5x
y = 10x
Explanation
For $y = kx$, use the point $(x, y) = (4, 10)$ to find $k$: $k = \frac{y}{x} = \frac{10}{4} = 2.5$. So the equation is $y = 2.5x$. A proportional graph must pass through the origin; equations with added constants (like $y = 4x + 10$) are not proportional.