Find k so that the following equations are consistent:
`2x-y+3=0, kx-y+1=0, 5x-y-3=0`

Options

A. 4
B. 3
C. 5
D. 9

Answer

To find the value of \( k \) such that the equations are consistent, we need to ensure that all three equations intersect at a single point. The equations given are:

1. \( 2x - y + 3 = 0 \) (Equation 1)
2. \( kx - y + 1 = 0 \) (Equation 2)
3. \( 5x - y - 3 = 0 \) (Equation 3)

### Step 1: Solve Equations 1 and 3

We will first solve Equations 1 and 3 to find their point of intersection.

From Equation 1:
\[ y = 2x + 3 \]

From Equation 3:
\[ y = 5x - 3 \]

### Step 2: Set the equations for \( y \) equal to each other

Since both expressions equal \( y \), we can set them equal to find \( x \):
\[ 2x + 3 = 5x - 3 \]

### Step 3: Rearrange the equation

Rearranging gives:
\[ 3 + 3 = 5x - 2x \]
\[ 6 = 3x \]

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

Dividing both sides by 3:
\[ x = 2 \]

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

Now substitute \( x = 2 \) back into Equation 1 to find \( y \):
\[ y = 2(2) + 3 = 4 + 3 = 7 \]

Thus, the point of intersection of Equations 1 and 3 is \( (2, 7) \).

### Step 6: Substitute \( x \) and \( y \) into Equation 2

Now we need to ensure that Equation 2 also passes through the point \( (2, 7) \):
\[ k(2) - 7 + 1 = 0 \]
This simplifies to:
\[ 2k - 6 = 0 \]

### Step 7: Solve for \( k \)

Now, solve for \( k \):
\[ 2k = 6 \]
\[ k = 3 \]

### Conclusion

Thus, the value of \( k \) that makes the equations consistent is:
\[ \boxed{3} \]