How do you Solve systems of equations?

Answer 1

When given multiple linear equations and asked to solve for certain variables, the first requirement is that you have as many equations as you have variables. If you have more equations than variables, that's fine; just use whichever are the simplest. If you have more variables than equations, the best you can do is solve for one variable in terms of another.

In a scholastic setting, you would normally have the same number of variables as equations. That is what will be assumed for the remainder of this article. The most common format of a system of linear equations is this:

y=2x+3

y=x+5

When asked to solve these equations, what's really being asked is for you to find where those two lines intersect. This allows you to assume that the x value and y value will be the same at that point. Therefore, you can say that

y=2x+3=x+5

2x+3=x+5

x=2

Then, since we know that x=2, we can put that value into either of the initial equations to solve for y. Let's choose the second one, because then we don't have to worry about multiplying by two.

y=x+5

y=2+5

y=7

Therefore, the two lines intersect at the coordinates (2,7). This method is normally referred to as solving by comparison, which is a special case of solving by substitution (see below).

Sometimes, equations are not formatted in a way that solving by comparison is convenient. For example, perhaps something like this is given:

6=9y-3x

2y=3x+4

In situations like these, it is often necessary to do some algebra before combining the equations in some way. Let's try to isolate the variable x in the first equation.

6=9y-3x

3x+6=9y

3x=9y-6

x=3y-2

Now that we have x in terms of y, let's put that new equation into the second given equation.

2y=3(3y-2)+4

2y=9y-6+4

2y=9y-2

2=7y

y=2/7

Again, putting this y value into the first equation, we get:

6=9y-3x

6=9(2/7)-3x

6=18/7-3x

6-18/7=3x

42/7-18/7=3x (in this step, we multiplied 6 by 7/7)

24/7=3x

24/7*(1/3)=x

x=24/21

Therefore the coordinates that these two lines intersect at is (24/21,2/7).

You'll notice that this set of coordinates are less appealing than the integers found in the first problem. Since substitution is a very general method of solving linear equations, it will work under any circumstances. If you have doubts about using comparison (or the final method, outlined below) then use substitution. Provided you don't make any mistakes in the mechanics, substitution will get you the correct answer.

The final method is called elimination. Using this method, one can often avoid long and tedious algebra. For example:

2x+y=1

-2x+2y=5

If we were to use substitution, we would get the correct answer (though we'd have to isolate a variable, expand a bracket, isolate for the other variable in terms of the first, substitute that equation into the other, and eventually solve for both variables). However, there is an easier way. We can add the two equations together.

2x+(-2x)+y+2y=1+5

0+3y=6

y=2

Substituting y=2 into the first equation, we get:

2x+y=1

2x+2=1

2x=-1

x=-(1/2)

Therefore the coordinates that these two lines intersect at is (-1/2,2)

It is worth noting that we are not limited to adding the two equations together. We may also subtract, multiply, and divide the equations. Elimination often takes the most practice to spot when it will be useful, but it can save many lines of math when applied in a suitable manner (as in the last example).

Those are the basic methods for solving two equation, two unknown problems. They will also hold true for three or more equation/unknown problems, but the mechanics of the solution becomes very long, very quickly. In more than two equation/variable questions, it is recommended to use matrices, but that is beyond the scope of this answer.