Question

The straight line L₁ passes through the points (–1, 3) and (11, 12).

a.   Find an equation for L₁ in the form ax + by + c = 0, where a, b and c are integers.

The line L₂ has equation 3y + 4x – 30 = 0.

b.   Find the coordinates of the point of intersection of L₁ and L₂ .

Solution

a.    

We are given that the straight line L₁ passes through the points A(–1, 3) and B(11, 12).

We are required to find the equation of the line L1.

Two-Point form of the equation of the line is;

Therefore;

b.
 

We are given that equation of the line L₂;

We are required to find the coordinates of point(s) of intersection of the lines L₁ and L₂.

If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e.  coordinates of that point have same values on both lines (or on the line and the curve).  Therefore, we can equate  coordinates of both lines i.e. equate equations of both the lines (or the  line and the curve).

Equation of the line L₂ is;

Equation of the line L₁ is;

We need to solve these simultaneous equations.

Multiply first equation with 4.

Multiply 2nd equation with 3.

Add the two equations.

Single value of x indicates that there is only one intersection point.

With x-coordinate of point of intersection of two lines (or line and the curve) at hand, we can find the  y-coordinate of the point of intersection of two lines (or line and the curve) by substituting value  of x-coordinate of the point of intersection in any of the two equations.

We choose equation of the line L₂;

Substitute;

Hence, coordinates of point of intersection of lines L₁ and L₂ are (3,6).