Represent the following system of linear equations graphically. From the graph, find the points where the lines intersect y-axis.
`3x+y -6=0`
`2x-y -5=0`

Answer

To solve the problem of representing the given system of linear equations graphically and finding the points where the lines intersect the y-axis, we can follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
We start with the two equations:
1. \(3x + y - 6 = 0\)
2. \(2x - y - 5 = 0\)

We can rearrange these equations to express \(y\) in terms of \(x\).

For the first equation:
\[ y = 6 - 3x \quad \text{(Equation 1)} \]

For the second equation:
\[ y = 2x - 5 \quad \text{(Equation 2)} \]

### Step 2: Find points for each equation
Next, we will find points for each equation by substituting different values of \(x\).

**For Equation 1: \(y = 6 - 3x\)**
- When \(x = 0\):
\[ y = 6 - 3(0) = 6 \quad \Rightarrow (0, 6) \]
- When \(x = 1\):
\[ y = 6 - 3(1) = 3 \quad \Rightarrow (1, 3) \]
- When \(x = 2\):
\[ y = 6 - 3(2) = 0 \quad \Rightarrow (2, 0) \]

**For Equation 2: \(y = 2x - 5\)**
- When \(x = 0\):
\[ y = 2(0) - 5 = -5 \quad \Rightarrow (0, -5) \]
- When \(x = 1\):
\[ y = 2(1) - 5 = -3 \quad \Rightarrow (1, -3) \]
- When \(x = 2\):
\[ y = 2(2) - 5 = -1 \quad \Rightarrow (2, -1) \]

### Step 3: Plot the points on a graph
Now we can plot the points we found on a graph:
- For Equation 1: Points (0, 6), (1, 3), and (2, 0)
- For Equation 2: Points (0, -5), (1, -3), and (2, -1)

### Step 4: Draw the lines
Connect the points for each equation to form straight lines.

### Step 5: Identify the intersection points with the y-axis
From the graph, we can observe where each line intersects the y-axis:
- The line from Equation 1 intersects the y-axis at the point (0, 6).
- The line from Equation 2 intersects the y-axis at the point (0, -5).

### Final Answer
The points where the lines intersect the y-axis are:
- For \(3x + y - 6 = 0\): (0, 6)
- For \(2x - y - 5 = 0\): (0, -5)