Section 5-3

Solving Linear Systems by the Linear Combination Method

(Also known as the Elimination Method or the Addition Method)

A system of linear equations is just a set of two or more linear equations.

In two variables, the graph of a system of two equations is a pair of lines in the plane. These lines will intersect in either zero points (if they are parallel), one point (in most cases), or infinitely many points (if the graph of the two equations is the same line).

If the lines intersect in one point, the Linear Combination Method will give you the point of intersection.

What is a Linear Combination?

If you have a true equation, you can add the same thing to both sides and it will still be true. For instance:

if x = y,

then x + 3 = y + 3.

So, if you have two true equations, you can add them together and still have a true equation:

if x = y and p = q,

then x + p = y + q,

because you've added an equal quantity to both sides.

Also, you can multiply both sides of any equation and it will still be true. That is

if x = y, then ax = ay.

A linear combination is a sum of multiples of linear equations. That means: start with a couple of linear equations, multiply them by some numbers, and then add them together. You will get a new equation which is a linear combination of the first two. (Linear combinations are a VERY important concept in mathematics, used by scientists and engineers in just about every field you can think of.)

Multiply the equation x + y = 1 by –2. We get:

–2x – 2y = –2

Now add this to the equation 2x + 5y = 8.

2x + 5y – 2x – 2y = 8 + (–2)

Simplify.

3y = 6

How do I use Linear Combinations to solve systems of equations?

Notice how in the last example, we ended up with an equation that only had y's (no x's). We eliminated the x (that's why this is also called the elimination method. You can now solve for y:

3y/3 = 6/3

y = 2

Now substitute in one of the original equations (say, x + y = 1), to find x.

x + 2 = 1

x = –1

So the intersection of the lines x + y = 1 and 2x + 5y = 8 is the point (–1, 2).

What does it mean if I get a false equation?

In some cases, you may get a false equation. For example, if you try to add the two lines:

x – y = 1

y = x

you get:

x = x + 1

Subtracting x from both sides, we have:

0 = 1

which is not true.

What happened? The two lines we started with were parallel. So there is no point of intersection, and no solution to the system.