Draw the graph of the equation
`4x+3y+6=0`
From the graph find
(i) `y_(1)` the value of y, when x=12
(ii) `y_(2)` the value of y, when x=-6

Answer

To solve the problem step by step, we will first rewrite the equation and then find the intercepts to plot the graph. Finally, we will substitute the given values of x to find the corresponding values of y.

### Step 1: Rewrite the Equation
The given equation is:
\[ 4x + 3y + 6 = 0 \]

### Step 2: Find the y-intercept
To find the y-intercept, set \( x = 0 \):
\[ 4(0) + 3y + 6 = 0 \]
\[ 3y + 6 = 0 \]
\[ 3y = -6 \]
\[ y = -2 \]
So, the y-intercept is \( (0, -2) \).

### Step 3: Find the x-intercept
To find the x-intercept, set \( y = 0 \):
\[ 4x + 3(0) + 6 = 0 \]
\[ 4x + 6 = 0 \]
\[ 4x = -6 \]
\[ x = -\frac{3}{2} \]
So, the x-intercept is \( \left(-\frac{3}{2}, 0\right) \).

### Step 4: Plot the Points
Now we have two points to plot on the graph:
1. \( (0, -2) \)
2. \( \left(-\frac{3}{2}, 0\right) \)

### Step 5: Draw the Graph
Draw a straight line through the points \( (0, -2) \) and \( \left(-\frac{3}{2}, 0\right) \). This line represents the equation \( 4x + 3y + 6 = 0 \).

### Step 6: Find \( y_1 \) when \( x = 12 \)
Substituting \( x = 12 \) into the equation:
\[ 4(12) + 3y + 6 = 0 \]
\[ 48 + 3y + 6 = 0 \]
\[ 3y + 54 = 0 \]
\[ 3y = -54 \]
\[ y = -\frac{54}{3} \]
\[ y = -18 \]
Thus, \( y_1 = -18 \).

### Step 7: Find \( y_2 \) when \( x = -6 \)
Substituting \( x = -6 \) into the equation:
\[ 4(-6) + 3y + 6 = 0 \]
\[ -24 + 3y + 6 = 0 \]
\[ 3y - 18 = 0 \]
\[ 3y = 18 \]
\[ y = \frac{18}{3} \]
\[ y = 6 \]
Thus, \( y_2 = 6 \).

### Final Answers:
(i) \( y_1 = -18 \) when \( x = 12 \)
(ii) \( y_2 = 6 \) when \( x = -6 \)