The equation of the line passing through (0,0) and intersection point of 3x-4y=2 and x+2y=-4 is

Options

A. 7x = 6y
B. 6x=7y
C. 5x=8y
D. x=0

Explanation

To find the equation of the line passing through the origin (0,0) and the intersection point of the lines given by the equations \(3x - 4y = 2\) and \(x + 2y = -4\), we can follow these steps:

### Step 1: Find the Intersection Point of the Two Lines

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

To find the intersection point, we can use the elimination method. First, we can multiply Equation 2 by 2 to make the coefficients of \(y\) easier to eliminate:

\[ 2(x + 2y) = 2(-4) \implies 2x + 4y = -8 \quad \text{(Equation 3)} \]

### Step 2: Add the Equations

Now we will add Equation 1 and Equation 3:

\[ (3x - 4y) + (2x + 4y) = 2 - 8 \]

This simplifies to:

\[ 5x = -6 \]

### Step 3: Solve for \(x\)

Now, we can solve for \(x\):

\[ x = -\frac{6}{5} \]

### Step 4: Substitute \(x\) Back to Find \(y\)

Next, we substitute \(x = -\frac{6}{5}\) back into either of the original equations to find \(y\). We can use Equation 2:

\[ -\frac{6}{5} + 2y = -4 \]

Rearranging gives:

\[ 2y = -4 + \frac{6}{5} \]

Finding a common denominator (which is 5):

\[ 2y = -\frac{20}{5} + \frac{6}{5} = -\frac{14}{5} \]

Now, divide by 2:

\[ y = -\frac{14}{10} = -\frac{7}{5} \]

### Step 5: Intersection Point

Thus, the intersection point of the two lines is:

\[ \left(-\frac{6}{5}, -\frac{7}{5}\right) \]

### Step 6: Equation of the Line through (0,0) and the Intersection Point

Now we need to find the equation of the line that passes through the points \((0,0)\) and \(\left(-\frac{6}{5}, -\frac{7}{5}\right)\).

Using the two-point form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]

Where \(m\) is the slope given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-\frac{7}{5} - 0}{-\frac{6}{5} - 0} = \frac{-\frac{7}{5}}{-\frac{6}{5}} = \frac{7}{6} \]

Substituting \(x_1 = 0, y_1 = 0\) into the equation gives:

\[ y - 0 = \frac{7}{6}(x - 0) \]

Thus, the equation simplifies to:

\[ y = \frac{7}{6}x \]

### Final Answer

The equation of the line passing through the origin and the intersection point is:

\[ 7x - 6y = 0 \]