Draw the graph of the equation, `2x-3y=5`.
From your graph , find (i) the value of y when x=4 and (ii) the value of x when y=3.

Answer

To solve the problem of drawing the graph of the equation \(2x - 3y = 5\) and finding specific values, we will follow these steps:

### Step 1: Rewrite the Equation in Slope-Intercept Form
We start by rewriting the equation in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y-intercept).

Starting with:
\[ 2x - 3y = 5 \]
We can rearrange it to solve for \(y\):
\[ -3y = -2x + 5 \]
\[ y = \frac{2}{3}x - \frac{5}{3} \]

### Step 2: Find the x-intercept and y-intercept
To draw the graph, we need two points: the x-intercept and the y-intercept.

**Finding the y-intercept** (set \(x = 0\)):
\[ y = \frac{2}{3}(0) - \frac{5}{3} = -\frac{5}{3} \]
So, the y-intercept is \((0, -\frac{5}{3})\).

**Finding the x-intercept** (set \(y = 0\)):
\[ 0 = \frac{2}{3}x - \frac{5}{3} \]
\[ \frac{2}{3}x = \frac{5}{3} \]
\[ x = \frac{5}{2} \]
So, the x-intercept is \((\frac{5}{2}, 0)\).

### Step 3: Plot the Points
Now we plot the points \((0, -\frac{5}{3})\) and \((\frac{5}{2}, 0)\) on the graph.

### Step 4: Draw the Line
Using a ruler, we draw a straight line through the two points plotted. This line represents the equation \(2x - 3y = 5\).

### Step 5: Find the Value of y when x = 4
Now, we will find the value of \(y\) when \(x = 4\):
Substituting \(x = 4\) into the original equation:
\[ 2(4) - 3y = 5 \]
\[ 8 - 3y = 5 \]
\[ -3y = 5 - 8 \]
\[ -3y = -3 \]
\[ y = 1 \]
So, when \(x = 4\), \(y = 1\). The point is \((4, 1)\).

### Step 6: Find the Value of x when y = 3
Next, we will find the value of \(x\) when \(y = 3\):
Substituting \(y = 3\) into the original equation:
\[ 2x - 3(3) = 5 \]
\[ 2x - 9 = 5 \]
\[ 2x = 5 + 9 \]
\[ 2x = 14 \]
\[ x = 7 \]
So, when \(y = 3\), \(x = 7\). The point is \((7, 3)\).

### Summary of Results
- The graph of the equation \(2x - 3y = 5\) has been drawn.
- The value of \(y\) when \(x = 4\) is \(1\).
- The value of \(x\) when \(y = 3\) is \(7\).