Question

The amount it costs (c) to fill a tank of gas is directly related to the number of gallons put in (g). Which of the following equations can be used to find the constant of variation (m)?

M/g = c
g/c = m
c/g = m
m/c = g

Answer 1

The correct equation to find the constant of variation (m) in this scenario is:
`\( c/g = m \).`

Explanation:

In the context of direct variation, the relationship between the two variables (cost, (c), and gallons, (g)) is expressed as ( c = mg), where ( m) is the constant of variation. To isolate ( m ) in this equation, we can rearrange it algebraically.

Dividing both sides of the equation by ( g ) gives:

`\[ (c)/(g) = (mg)/(g) \]`

Simplifying the right side by canceling out the common factor of ( g ) leaves us with the desired equation:

`\[ (c)/(g) = m \]`

This equation signifies that the constant of variation (( m )) is equal to the ratio of cost to gallons. In other words, it tells us how much cost changes for each additional gallon of gas. Therefore, the correct expression to find ( m) is ( c/g = m ).

Understanding the relationship between variables and the proper manipulation of equations is crucial in solving problems involving direct variation. In this case, recognizing that the constant of variation is obtained by dividing the cost by the number of gallons allows for an accurate and meaningful interpretation of the equation.

Answer 2

g/c = m The equation showcases the direct relationship between the variables, outlining that the constant of variation remains the same regardless of the specific values of gallons or cost involved. Understanding this relationship helps in predicting costs based on the quantity of gallons filled.

Explanation:

The equation g/c = m can be used to find the constant of variation (m) in this scenario. In a direct variation equation, the constant of variation (m) represents the ratio between two variables, in this case, gallons (g) and cost (c).

The equation g/c = m directly expresses that the number of gallons (g) divided by the cost (c) is equal to the constant of variation (m). This signifies that as the number of gallons increases, the cost also increases proportionally, maintaining a constant ratio between the two values.

The equation showcases the direct relationship between the variables, outlining that the constant of variation remains the same regardless of the specific values of gallons or cost involved. Understanding this relationship helps in predicting costs based on the quantity of gallons filled.