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.

Answer Details

When a variable \( y \) is "partly constant and partly varies as \( x \)", it means:
There is a fixed (constant) part and a part that changes directly with \( x \). This can be written as: \[ y = a + bx \] where:

• \( a \) is the constant part (it does not depend on \( x \)),
• \( b \) is the coefficient showing how \( y \) changes as \( x \) changes.

We are given two pieces of information:

• When \( x = 3 \), \( y = 7 \)
• When \( x = 5 \), \( y = 11 \)

Plug these values into the equation \( y = a + bx \):

• When \( x = 3 \), \( y = 7 \): \[ 7 = a + 3b \]
• When \( x = 5 \), \( y = 11 \): \[ 11 = a + 5b \]

Now solve these two equations simultaneously.

Step 1: Subtract the first equation from the second:

\[ (11 = a + 5b) - (7 = a + 3b) \] \[ 11 - 7 = (a + 5b) - (a + 3b) \] \[ 4 = 2b \]

So, \[ b = \frac{4}{2} = 2 \]

Step 2: Put \( b = 2 \) back into either equation to find \( a \). Let's use \( 7 = a + 3b \):

\[ 7 = a + 3 \times 2 \] \[ 7 = a + 6 \] \[ a = 7 - 6 = 1 \]

So the constants of variation are:

• \( a = 1 \) (the constant part)
• \( b = 2 \) (the coefficient of \( x \))

The relationship is: \[ y = 1 + 2x \]

The correct constants are 1 and 2.