Find the equation of line joining the origin to the point of intersection of `4x+3y=8 and x+y=1`.

Answer

To find the equation of the line joining the origin to the point of intersection of the lines given by the equations \(4x + 3y = 8\) and \(x + y = 1\), we can follow these steps:

### Step 1: Find the point of intersection of the two lines.

We have the two equations:
1. \(4x + 3y = 8\) (Equation 1)
2. \(x + y = 1\) (Equation 2)

To find the intersection, we can solve these equations simultaneously. We can express \(y\) from Equation 2:

\[ y = 1 - x \]

### Step 2: Substitute \(y\) in Equation 1.

Now, substitute \(y\) in Equation 1:

\[ 4x + 3(1 - x) = 8 \]

### Step 3: Simplify the equation.

Expanding the equation gives:

\[ 4x + 3 - 3x = 8 \]

Combine like terms:

\[ (4x - 3x) + 3 = 8 \]

This simplifies to:

\[ x + 3 = 8 \]

### Step 4: Solve for \(x\).

Subtracting 3 from both sides gives:

\[ x = 5 \]

### Step 5: Substitute \(x\) back to find \(y\).

Now, substitute \(x = 5\) back into Equation 2 to find \(y\):

\[ y = 1 - 5 = -4 \]

### Step 6: Identify the point of intersection.

Thus, the point of intersection is \((5, -4)\).

### Step 7: Find the slope of the line joining the origin to the point of intersection.

The slope \(m\) of the line passing through the origin \((0, 0)\) and the point \((5, -4)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{5 - 0} = \frac{-4}{5} \]

### Step 8: Write the equation of the line.

The equation of the line in slope-intercept form is given by:

\[ y = mx \]

Substituting \(m = -\frac{4}{5}\):

\[ y = -\frac{4}{5}x \]

### Step 9: Convert to standard form.

To convert this into standard form \(Ax + By + C = 0\):

\[ 4x + 5y = 0 \]

Thus, the required equation of the line joining the origin to the point of intersection is:

\[ 4x + 5y = 0 \]

---