Find the equation of line joining origin to the point of intersection of the pair of lines `3x+y=10` and `x-y=2`.

Answer

To find the equation of the line joining the origin to the point of intersection of the lines given by the equations \(3x + y = 10\) and \(x - y = 2\), we can follow these steps:

### Step 1: Solve the system of equations
We have two equations:
1. \(3x + y = 10\) (Equation 1)
2. \(x - y = 2\) (Equation 2)

To find the point of intersection, we can solve these equations simultaneously.

### Step 2: Express \(y\) from Equation 2
From Equation 2, we can express \(y\) in terms of \(x\):
\[ y = x - 2 \]

### Step 3: Substitute \(y\) in Equation 1
Now substitute \(y\) from Equation 2 into Equation 1:
\[ 3x + (x - 2) = 10 \]
This simplifies to:
\[ 3x + x - 2 = 10 \]
\[ 4x - 2 = 10 \]

### Step 4: Solve for \(x\)
Add 2 to both sides:
\[ 4x = 12 \]
Now divide by 4:
\[ x = 3 \]

### Step 5: Find \(y\) using the value of \(x\)
Substitute \(x = 3\) back into the expression for \(y\):
\[ y = 3 - 2 = 1 \]

### Step 6: Identify the point of intersection
The point of intersection of the two lines is \((3, 1)\).

### Step 7: Find the slope of the line from the origin to the point of intersection
The origin is \((0, 0)\) and the point of intersection is \((3, 1)\). The slope \(m\) of the line joining these two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{3 - 0} = \frac{1}{3} \]

### Step 8: Write the equation of the line
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = \frac{1}{3}\), \(x_1 = 0\), and \(y_1 = 0\):
\[ y - 0 = \frac{1}{3}(x - 0) \]
This simplifies to:
\[ y = \frac{1}{3}x \]

### Final Answer
The equation of the line joining the origin to the point of intersection is:
\[ y = \frac{1}{3}x \]