Equation Of A Line

Step 1: The given equation is , which means the x-coordinate of every point on the line is fixed at 7.

Step 2: Such an equation represents a vertical line that runs up and down through all values of , but always where .

Step 3: A vertical line is parallel to the y-axis, because both extend infinitely in the vertical direction and never intersect.

Step 4: This line does not pass through the origin , since all its points have , not .

Step 5: It also does not pass through the point , because this point lies on the vertical line , while our line is at .

Conclusion: The line is parallel to the y-axis.

Step 1: Write the given equations

Step 2: Introduce substitutions

Let and

Then equations (1) and (2) become:

Step 3: Eliminate one variable

To eliminate , multiply equation (3) by 4 and equation (4) by 3:

Equation (3) × 4:

Equation (4) × 3:

Step 4: Add equations (5) and (6)

So,

Step 5: Substitute the value of into equation (3) to find

Substitute into :

Step 6: Find the values of and

Since , then
Since , then

The correct option is (C):

Step 1: To determine if a line passes through the origin , we substitute into the equation and check if .

Step 2: Let's evaluate each option:
Option (1):
Put :
Hence, the line passes through the origin. ✅

Option (2):
Put :
Hence, the line passes through the point , not the origin. ❌

Option (3):
Put :
Hence, the line passes through , not the origin. ❌

Option (4):
This represents a horizontal line at height , and it passes through the origin only when , which is not guaranteed. ❌

Step 3: Therefore, the only line that definitely passes through is: