Elimination

Solving a System of Two Equations Graphically

The solutions to a single linear equation are the points on its graph, which is a straight line. For a point to represent a solution to two linear equations, it must lie simultaneously on both of the corresponding lines. In other words, it must be a point where the two lines cross, or intersect.
Thus, to locate solutions to a system of two equations in two unknows, plot the graphs, and locate the intersection points (if any).

Solving a System of Two Equations in Two Unknowns by Elimination

Q Do we really need another method of solving a system of linear equations? A The problem with the graphical approach is that it only gives approximate solutions; locating the exact point of intersection of two lines would require perfect accuracy, which is impossible in practice.

The method of elimination is an algebraic way of obtaining the exact solution(s) of a system of equations in two unknowns by manipulating the equations in such a way as to eliminate of the variables (x or y). The best way to understand this method is through some examples.

Examples
(a) Solve
2x + 3y = 4 x - 3y = 2
If we simply add these equations (add the left-hand sides and the right-hand sides) the y's cancel out, and we get
3x = 6, giving x = 2.
To obtain y, we substitute x = 2 in either of the two equations (let us choose the first):
2(2) + 3y = 4, giving 4 + 3y = 4, so that 3y = 0, or y = 0.
Thus, the solution is (x, y) = (2, 0).

(b) Solve
2x + 3y = 3 3x - 2y = -2
This time, adding (or subtracting) the equations does not result in either x or y being eliminated. However, we can eliminiate x by multiplying the first equation by 3 and the second by -2:
2x + 3y = 3
3
6x + 9y = 9
3x - 2y = -2
-2
-6x + 4y = 4

Now if we add them, we get
13y = 13, giving y = 1
To obtain x, we substitute y = 1 in either of the two equations (let us choose the first):
2x + 3(1) = 3, giving 2x + 3 = 3, so x = 0
Thus, the solution is (x, y) = (0, 1).