Using a scale of 1cm to 1unit for both the axes Draw the graph of the following equations : `6y=5x+10, y=5x-15` From the graph find:
The co-ordinates of the points where the two lines intersect.

Answer

To solve the problem of graphing the equations \(6y = 5x + 10\) and \(y = 5x - 15\) and finding their intersection point, we can follow these steps:

### Step 1: Rearranging the Equations
First, we need to rearrange the first equation into the slope-intercept form \(y = mx + b\).

1. Start with the equation:
\[ 6y = 5x + 10 \]
2. Divide every term by 6:
\[ y = \frac{5}{6}x + \frac{10}{6} \]
3. Simplify \(\frac{10}{6}\) to \(\frac{5}{3}\):
\[ y = \frac{5}{6}x + \frac{5}{3} \]

The second equation is already in slope-intercept form:
\[ y = 5x - 15 \]

### Step 2: Finding Points for the First Equation
We will find two points for the first equation \(y = \frac{5}{6}x + \frac{5}{3}\).

1. Let \(x = 0\):
\[ y = \frac{5}{6}(0) + \frac{5}{3} = \frac{5}{3} \]
Point: \((0, \frac{5}{3})\)

2. Let \(y = 0\):
\[ 0 = \frac{5}{6}x + \frac{5}{3} \]
Rearranging gives:
\[ \frac{5}{6}x = -\frac{5}{3} \]
Multiplying both sides by \(\frac{6}{5}\):
\[ x = -2 \]
Point: \((-2, 0)\)

### Step 3: Finding Points for the Second Equation
Now we will find two points for the second equation \(y = 5x - 15\).

1. Let \(x = 4\):
\[ y = 5(4) - 15 = 20 - 15 = 5 \]
Point: \((4, 5)\)

2. Let \(y = 0\):
\[ 0 = 5x - 15 \]
Rearranging gives:
\[ 5x = 15 \implies x = 3 \]
Point: \((3, 0)\)

### Step 4: Plotting the Points
Now we plot the points on a graph using a scale of 1 cm = 1 unit.

- For the first equation \(y = \frac{5}{6}x + \frac{5}{3}\):
- Plot the points \((0, \frac{5}{3})\) and \((-2, 0)\).

- For the second equation \(y = 5x - 15\):
- Plot the points \((4, 5)\) and \((3, 0)\).

### Step 5: Drawing the Lines
Draw straight lines through the points plotted for each equation.

### Step 6: Finding the Intersection
Look for the point where the two lines intersect. From the graph, estimate the coordinates of the intersection point.

### Conclusion
From the graph, the intersection point is approximately \((4.2, 5.5)\).