Y is partly constant and partly varies as x. When x = 3, y = 7 and when x = 5, y = 11. Find the constants of variation.

Analysis

The problem describes a relationship where $y$ is the sum of a constant and a variable part that changes with $x$. Mathematically, this is expressed as: $y = k + cx$ Where:

• $k$ is the constant.
• $c$ is the constant of variation (the rate at which $y$ changes with $x$).

Concept

To solve this, we need to create a system of linear equations using the provided pairs of values $(x, y)$ and solve for the two unknowns, $k$ and $c$.

Step-by-Step Solution

Step 1: Set up the equations based on the given data.

• When $x = 3, y = 7$: $7 = k + 3c$ --- (Equation 1)
• When $x = 5, y = 11$: $11 = k + 5c$ --- (Equation 2)

Step 2: Solve the system of equations. We can eliminate $k$ by subtracting Equation 1 from Equation 2: $(k + 5c) - (k + 3c) = 11 - 7$ $k - k + 5c - 3c = 4$ $2c = 4$ $c = \frac{4}{2}$ $c = 2$

Step 3: Substitute the value of $c$ back into Equation 1 to find $k$. $7 = k + 3(2)$ $7 = k + 6$ $k = 7 - 6$ $k = 1$

The constants are $k = 1$ and $c = 2$.

Conclusion

The constants of variation are 1 and 2. Comparing this to your options, this corresponds to option B.

Exam Tip

When you see the phrase "partly constant and partly varies as x," always write down $y = k + cx$ immediately. If the question had said "partly constant and partly varies as the square of x," your equation would have been $y = k + cx^2$. Always look for those keywords!