Question

SOLVE EACH EQUATION

The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x=3

1. x= 1, y=9

2. x= -3, y=8

3. x=7, y=2

4. x=3/4, y=28

5. x= -4, y = - 5/4

6. x= -12, y= - 1/6

Answer 1

For inverse variation, we can express the relationship between \(x\) and \(y\) as:

\[
xy = k
\]

where \(k\) is a constant. We will first find \(k\) using the given values of \(x\) and \(y\), and then we will use this constant to find \(y\) when \(x = 3\).

1. **For \(x = 1, y = 9\)**:
\[
k = xy = 1 \cdot 9 = 9
\]
The equation is \(xy = 9\).
To find \(y\) when \(x = 3\):
\[
3y = 9 \quad \Rightarrow \quad y = \frac{9}{3} = 3
\]

2. **For \(x = -3, y = 8\)**:
\[
k = xy = -3 \cdot 8 = -24
\]
The equation is \(xy = -24\).
To find \(y\) when \(x = 3\):
\[
3y = -24 \quad \Rightarrow \quad y = \frac{-24}{3} = -8
\]

3. **For \(x = 7, y = 2\)**:
\[
k = xy = 7 \cdot 2 = 14
\]
The equation is \(xy = 14\).
To find \(y\) when \(x = 3\):
\[
3y = 14 \quad \Rightarrow \quad y = \frac{14}{3} \approx 4.67
\]

4. **For \(x = \frac{3}{4}, y = 28\)**:
\[
k = xy = \frac{3}{4} \cdot 28 = 21
\]
The equation is \(xy = 21\).
To find \(y\) when \(x = 3\):
\[
3y = 21 \quad \Rightarrow \quad y = \frac{21}{3} = 7
\]

5. **For \(x = -4, y = -\frac{5}{4}\)**:
\[
k = xy = -4 \cdot -\frac{5}{4} = 5
\]
The equation is \(xy = 5\).
To find \(y\) when \(x = 3\):
\[
3y = 5 \quad \Rightarrow \quad y = \frac{5}{3} \approx 1.67
\]

6. **For \(x = -12, y = -\frac{1}{6}\)**:
\[
k = xy = -12 \cdot -\frac{1}{6} = 2
\]
The equation is \(xy = 2\).
To find \(y\) when \(x = 3\):
\[
3y = 2 \quad \Rightarrow \quad y = \frac{2}{3} \approx 0.67
\]

### Summary of results when \(x = 3\):
1. \( y = 3 \)
2. \( y = -8 \)
3. \( y \approx 4.67 \)
4. \( y = 7 \)
5. \( y \approx 1.67 \)
6. \( y \approx 0.67 \)