Question

How do you solve the system using the elimination method for -7x + 6y = -4 and 14x - 12y = 8?

Answer 1

`-7x + 6y = -4`

Explanation:

Firstly, let us understand what 'solving' means. When we solve a system of equations (of say two variables) for the unknowns (say `x` and `y`), we find (either one or many) point(s) `(x,y)` on the x-y plane which satisfies all the equations.

Secondly, note that if we multiply an equation by a constant term (this can be negative, or any real number), the equation remains the same. For example, if `x` and `y` represent the number of apples and oranges a person has, and the total number of fruits are 4, it means `x + y = 4`. But this is equivalent to saying that if he had twice the number of individual fruits, he would have twice the number of total fruits.

In the usual case, when you solve two linear equations, you get a single solution for `(x,y)`. This is because two non-parallel lines intersect at only one point in the 2-dimensional plane. We say that this problem has a unique solution. (Draw two non-parallel lines. They must intersect at some point.)

The fact is that, this is not the only case. There may be two parallel lines. In this particular case, they never intersect, and there is no solution. Again, draw and check.

The third case occurs when two lines overlap. In this case, all the points on either line satisfy both equations. This is the case of infinite solutions. We get a line instead of a point as the solution.

Now let us look at the problem you stated.
The first equation: `-7x + 6y = -4`
And the second: `14x - 12y = 8`

If we multiply the first equation by -2, we get the second equation. Thus they are identical, and we are in the 3rd case of overlapping equations. Thus there are infinite solutions. All the points satisfying the first equation are solutions to the system of equation. Thus our solution is `-7x + 6y = -4`.