# A Math Primer

**B. Addition Method
**

The whole problem with solving a system of

equations is that you cannot solve an equation that

has two unknowns in it. You need an equation with

only one variable so that you can isolate the variable

on one side of the equation. Both methods that we

will look at are techniques for eliminating one of the

variables to give you an equation in just one

unknown , which you can then solve by the usual

methods.

The first method of solving systems of linear

equations is the addition method, in which the two

equations are added together to eliminate one of the

variables.

Adding the equations means that we add the left

sides of the two equations together, and we add the

right sides together. This is legal because of the

Addition Principle, which says that we can add the

same amount to both sides of an equation. Since the

left and right sides of any equation are equal to each

other, we are indeed adding the same amount to both

sides of an equation.

Consider this simple example :

**Example:**

If we add these equations together, the terms

containing y will add up to zero (2y plus -2y), and we

will get

or

5x = 5

**x = 1
**

However, we are not finished yet—we know x, but

we still don’t know y. We can solve for y by

substituting the now known value for x into either of

our original equations. This will produce an equation

that can be solved for y:

Now that we know both x and y, we can say that the

solution to the system is the pair (1, 1/2).

This last example was easy to see because of the

fortunate presence of both a positive and a negative

2y. One is not always this lucky. Consider

**Example:**

Now there is nothing so obvious, but there is still

something we can do. If we multiply the first

equation by -3, we get

(Don’t forget to multiply every term in the equation,

on both sides of the equal sign ). Now if we add them

together the terms containing x will cancel:

or

As in the previous example, now that we know y we

can solve for x by substituting into either original

equation. The first equation looks like the easiest to

solve for x, so we will use it:

And so the solution point is (-4, 7/2).

Now we look at an even less obvious example:

**Example:**

Here there is nothing particularly attractive about

going after either the x or the y. In either case, both

equations will have to be multiplied by some factor

to arrive at a common coefficient . This is very much

like the situation you face trying to find a least

common denominator for adding fractions , except

that here we call it a Least Common Multiple

(LCM). As a general rule , it is easiest to eliminate

the variable with the smallest LCM. In this case that

would be the y, because the LCM of 2 and 3 is 6. If

we wanted to eliminate the x we would have to use

an LCM of 10 (5 times 2). So, we choose to make the

coefficients of y into plus and minus 6. To do this,

the first equation must be multiplied by 3, and the

second equation by 2:

or

Now adding these two together will eliminate the

terms containing y:

or

x = 2

We still need to substitute this value into one of the

original equation to solve for y:

Thus the solution is the point (2, 2).

**C. Substitution Method
**

When we used the Addition Method to solve a

system of equations, we still had to do a substitution

to solve for the remaining variable. With the

substitution method, we solve one of the equations

for one variable in terms of the other, and then

substitute that into the other equation. This makes

more sense with an example:

**Example:**

2y + x = 3 (1)

4y – 3x = 1 (2)

Equation 1 looks like it would be easy to solve for x,

so we take it and isolate x:

2y + x = 3

x = 3 – 2y (3)

Now we can use this result and substitute 3 - 2y in

for x in equation 2:

Now that we have y, we still need to substitute back

in to get x. We could substitute back into any of the

previous equations, but notice that equation 3 is

already conveniently solved for x:

And so the solution is (1, 1).

As a rule, the substitution method is easier and

quicker than the addition method when one of the

equations is very simple and can readily be solved

for one of the variables.

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