Question

If the given quantities are proportional, find the value of y when x = 10.

2
10
9
20

Answer 1

To determine the value of \( y \) when \( x = 10 \) given that the values are proportional, we can set up a relationship based on the values provided.

The quantities given are:
- Let \( x_1 = 2 \), \( y_1 = 10 \)
- Let \( x_2 = 9 \), \( y_2 = 20 \)

Since the quantities are proportional, we can express this relationship as:

\[
\frac{y_1}{x_1} = \frac{y_2}{x_2}
\]

Substituting the known values:

\[
\frac{10}{2} = \frac{20}{9}
\]

Calculating both sides gives:

\[
5 = \frac{20}{9}
\]

This equation does not hold true, meaning the assumption of direct proportionality for these values might not lead to a specific relationship.

If we need to find \( y \) when \( x = 10 \) with assumed values for \( x \) and \( y \):

Using the values given, we can try to find a mathematical relationship between \( y \) and \( x \) based on the proportionality seen:

1. For \( x=2 \), corresponding \( y=10 \).
2. For \( x=9 \), corresponding \( y=20 \).

### Finding the linear relationship

Let’s assume a linear relationship between \( x \) and \( y \). Observing the ratios:

\[
\text{From } (2, 10) \text{ to } (9, 20)
\]

The change in \( y \) as \( x \) increases:
- From 2 to 9, \( x \) increases by 7.
- From 10 to 20, \( y \) increases by 10.

The slope \( m \) can thus be calculated as:

\[
m = \frac{\Delta y}{\Delta x} = \frac{20 - 10}{9 - 2} = \frac{10}{7}
\]

Now, we can use point-slope form of a line \( y - y_1 = m(x - x_1) \) using one of the points, let's use (2, 10):

\[
y - 10 = \frac{10}{7}(x - 2)
\]

Now substituting \( x = 10 \):

\[
y - 10 = \frac{10}{7}(10 - 2) = \frac{10}{7} \cdot 8 = \frac{80}{7}
\]

Finally:

\[
y = 10 + \frac{80}{7} = \frac{70}{7} + \frac{80}{7} = \frac{150}{7} \approx 21.43
\]

Thus, the value of \( y \) when \( x = 10 \) is approximately **21.43**.