Draw the graph of the equation `3x+2y-5=0`. Use this graph to find:
(i) `x_(1)` the value of x, when y=7
(ii) `y_(1)` the value of y, when x=3

Answer

To solve the problem step-by-step, we will first draw the graph of the equation \(3x + 2y - 5 = 0\) and then use this graph to find the required values.

### Step 1: Rearranging the Equation
We start with the equation:
\[ 3x + 2y - 5 = 0 \]
Rearranging it to express \(y\) in terms of \(x\):
\[ 2y = -3x + 5 \]
\[ y = -\frac{3}{2}x + \frac{5}{2} \]

### Step 2: Finding Points to Plot
To draw the graph, we can find some points by substituting different values of \(x\) into the equation.

1. **When \(x = 0\)**:
\[ y = -\frac{3}{2}(0) + \frac{5}{2} = \frac{5}{2} = 2.5 \]
Point: \((0, 2.5)\)

2. **When \(x = 1\)**:
\[ y = -\frac{3}{2}(1) + \frac{5}{2} = -\frac{3}{2} + \frac{5}{2} = 1 \]
Point: \((1, 1)\)

3. **When \(x = 2\)**:
\[ y = -\frac{3}{2}(2) + \frac{5}{2} = -3 + \frac{5}{2} = -\frac{1}{2} \]
Point: \((2, -0.5)\)

4. **When \(x = 3\)**:
\[ y = -\frac{3}{2}(3) + \frac{5}{2} = -\frac{9}{2} + \frac{5}{2} = -2 \]
Point: \((3, -2)\)

### Step 3: Plotting the Points
Now we can plot the points \((0, 2.5)\), \((1, 1)\), \((2, -0.5)\), and \((3, -2)\) on a graph.

### Step 4: Drawing the Line
After plotting the points, we connect them with a straight line, which represents the equation \(3x + 2y - 5 = 0\).

### Step 5: Finding \(x_1\) when \(y = 7\)
To find \(x_1\) when \(y = 7\), we substitute \(y = 7\) into the original equation:
\[ 3x + 2(7) - 5 = 0 \]
\[ 3x + 14 - 5 = 0 \]
\[ 3x + 9 = 0 \]
\[ 3x = -9 \]
\[ x = -3 \]
Thus, \(x_1 = -3\).

### Step 6: Finding \(y_1\) when \(x = 3\)
To find \(y_1\) when \(x = 3\), we substitute \(x = 3\) into the original equation:
\[ 3(3) + 2y - 5 = 0 \]
\[ 9 + 2y - 5 = 0 \]
\[ 2y + 4 = 0 \]
\[ 2y = -4 \]
\[ y = -2 \]
Thus, \(y_1 = -2\).

### Summary of Results
- \(x_1\) when \(y = 7\) is \(-3\).
- \(y_1\) when \(x = 3\) is \(-2\).