Solving Systems by Elimination

OK, OK. Now this is the third way we can solve systems of equations. It’s exciting!!!

Remember how you can do ANYTHING you want (except divide by 0) to one side of an equation, as long as you do the same thing to the other side of the equation?

Well, we can use this trick to add two equations together.

Take, for example, the following system of equations:

3x – 4y = -2

2x + 4y = -3

If we take 3x – 4y = -2, we can add 2x + 4y to one side of the equation, and 3 to the other side because 2x + 4y and 3 are equal. So we’re actually adding the same thing to both sides!

We add straight down, combining like terms.

3x	-4y	=	-2
+2x	+4y		+(-3)
5x	+0	=	-5
5x		=	-5	Woah, check it out! One of our variables is gone! Now we can solve for x!
x		=	-1

Cool. Now to find y, we just substititute -1 for x in one of the equations.

3(-1)	-4y	=	-2	Simplify
-3	-4y	=	-2
+3			+3	Subtract 3 to get the y’s alone
	-4y	=	1	And divide by -4 to get the y by itself
	-4		-4
	y	=	-1 4	Hooray! We’re done!

So our solution is (-1, -1/4)

We can either add or subtract our two equations, depending upon what will cancel out one of the variables.

MULTIPLYING TO ELIMINATE

Now, let’s look at an example where it’s a little bit harder to solve by elimination.

3x + 5y = 7

4x + 10y = 16

Oh great, there are no coefficients that are the same! How will we ever cancel things out!?!

Luckily, we can use that same super-useful math fact, that we can do whatever we want to an equation as long as we do it to both sides.

Watch what happens if we multiply the first equation by -2…

-2(3x + 5y) = -2(7)

-6x – 10y = -14

Now the coefficients of y will cancel! We replace the first equation with the multiplied by -2 equation we just made and solve by elimination!

-6x	-10y	=	-14	Add down
+4x	+10y	=	+16	Combine like terms
-2x	+0	=	2
-2x		=	2	Hooray! Only one variable! Let’s solve!
x		=	-1

Now we substitute that x = -1 back into one of our original equations, and we get:

3(-1)	+ 5y	=	7	Simplify
-3	+ 5y	=	7
+3			+3	Solve for y by getting the y’s alone
	5y	=	10	Then dividing by 5 to get y by itself
	5		5
	y	=	2	Hooray! Now we know x and y!

So our solution is (-1, 2)

Systems of Equations and Solutions

SYSTEMS OF EQUATIONS AND SOLUTIONS

SYSTEM OF EQUATIONS

A “SYSTEM” is a set of equations with the same variables.

EX:

x - 3y = 12

2x + 4y = 4

We’re looking at systems of linear equations—equations of lines.

SOLUTIONS

The SOLUTION of a system of equations is made up of solutions that work for both equations.

EX: The solution to the equations above is x = 6, y = -2) because that point fits into both equations:

Plugging in 6 for x and -2 for y in the first equation, we get

6 – 3(-2) = 12

6 + 6 = 12

12 = 12

Plugging in 6 for x and -2 for y in the second equation, we get

2(6) + 4(-2) = 4

12 – 8 = 4

4 = 4

GRAPHING SYSTEMS

We can visualize systems by graphing the equations on the same set of axes. The solution is where the lines cross.

This is the graph of the example above. You can see the lines cross at (6, -2).

A system of equations may have ONE solution, MANY solutions, or NO SOLUTIONS.

If the lines cross ONCE, there is ONE specific solution.

If the lines are the SAME, they are crossing many, many times (at every single point on the line), so there are infinitely MANY SOLUTIONS.

If the lines are PARALLEL, there are NO SOLUTIONS, because the lines never cross.

(Note that these lines have the same slope but different intercepts.)

Graphing from Standard Form

STANDARD FORM

Standard form of the equation of a line looks like this:

Ax + By = C
We want A, B, and C to be whole numbers, and we want A to be positive.

We can graph from standard form by using the x- and y-intercepts.


INTERCEPTS

The x-intercept is where the line crosses the x-axis.
The y-intercept is where the line crosses the y-axis.

Note that at the x-intercept, y is zero. (We haven't moved up or down from the x-axis.)
At the y-intercept, x is zero. (We haven't moved right or left from the y-axis.)


GRAPHING FROM STANDARD FORM

Ex: 3x-4y=12

To find the x-intercept, we plug in 0 for y.

3x-4(0)-12 ..........................4(0) becomes 0
3x=12 ................................ now divide both sides by 3
x=4

So 4 is the x-intercept.

To find the y-intercept, we plug in zero for x.

3(0)-4x-12 ..........................3(0) becomes 0
-4x=12 ................................now divide both sides by -4
x=-3

So -3 is the y-intercept.

Graph the two intercepts and draw a line through them.