Find the equation of a line whose
`(i)` Slope `=-1` and `Y-` intercept `=3`.
`(ii)` Slope `=(2)/(5)` and `Y-` intercept `=-2`.
`(iii)` Slope `=(1)/(3)` and `Y-` intercept `=(2)/(3)`.

Answer

To find the equations of the lines based on the given slopes and y-intercepts, we can use the slope-intercept form of the equation of a line, which is:

\[ y = mx + c \]

where:
- \( m \) is the slope,
- \( c \) is the y-intercept.

Now, let's solve each part step by step.

### (i) Slope = -1 and Y-intercept = 3

1. **Identify the slope and y-intercept:**
- Slope \( m = -1 \)
- Y-intercept \( c = 3 \)

2. **Substitute the values into the equation:**
\[ y = mx + c \implies y = -1x + 3 \]

3. **Rearranging the equation:**
\[ y + x = 3 \implies x + y = 3 \]

**Final Equation for (i):**
\[ x + y = 3 \]

### (ii) Slope = \( \frac{2}{5} \) and Y-intercept = -2

1. **Identify the slope and y-intercept:**
- Slope \( m = \frac{2}{5} \)
- Y-intercept \( c = -2 \)

2. **Substitute the values into the equation:**
\[ y = mx + c \implies y = \frac{2}{5}x - 2 \]

3. **Multiply through by 5 to eliminate the fraction:**
\[ 5y = 2x - 10 \]

4. **Rearranging the equation:**
\[ 2x - 5y = 10 \]

**Final Equation for (ii):**
\[ 2x - 5y = 10 \]

### (iii) Slope = \( \frac{1}{3} \) and Y-intercept = \( \frac{2}{3} \)

1. **Identify the slope and y-intercept:**
- Slope \( m = \frac{1}{3} \)
- Y-intercept \( c = \frac{2}{3} \)

2. **Substitute the values into the equation:**
\[ y = mx + c \implies y = \frac{1}{3}x + \frac{2}{3} \]

3. **Multiply through by 3 to eliminate the fraction:**
\[ 3y = x + 2 \]

4. **Rearranging the equation:**
\[ x - 3y = -2 \]

**Final Equation for (iii):**
\[ x - 3y = -2 \]

### Summary of the Equations:
1. \( x + y = 3 \)
2. \( 2x - 5y = 10 \)
3. \( x - 3y = -2 \)