System of Equations!

By Christine y.

Let's get started!!

So you're probably wondering what the three different types of solutions are considered when solving a System of Equations, am I right? Huh? Huuh?


Well, here you go (look at the pictures):

Still confused? Here are some examples of what the 3 solutions look like when you try to solve it!

Graphing 500

Solve by Graphing
2x + 7y = 14
5x + 7y = ­7


Solution? This Video will show you.

Solving by Graphing

Sub-sub-substitution!! 500

Solve by Substitution.


3x ­- 5y = 13

x + 4y = 10


Solution? This Video will show you.

Solving with Substitution

Elimination!!! 500

Solve by Elimination.


3y = x + 5

­3x + 8y = 8


How to solve:



  1. First you would turn both equations into standard format.
  2. After that, you will have to choose a variable such as example, -1x and 3x or 8y and 3y.
  3. When you are done choosing the two variables, you will have to find the LCM of the two variables or the Least common Multiple of the 2. Example: 3x and -1x, the least common multiple of the 2 is -3.
  4. After that, both equations will have be multiplied by what ever number it takes for the variable (in this case 3x and 1x) to become that multiple or in this case 3. You may be confused so here is an example: (-1x+3y=5) (3) The whole equation will have to multiply by -1 because it is what number -1x has to multiply by to get the LCM of -3.
  5. You will have to do the same thing with 3x+8y=8
  6. After that, you will have to add both of the equations together. Example: -3x+9y=15 plus 3x+8y=8
  7. You should then get the equation of 17y=23
  8. Then you have to cancel out the 17 by dividing it on both sides.
  9. y=(23/17)



  1. Once you have the y variable figured out, you will have to plug in the number to one of the equations to get x. Example: 3y=x+5 --> 3*(23/17)=x+5
  2. You should then get the equation of (69/17)=x+5
  3. After that, you subtract 5 from both sides and should get the answer.
  4. x=(-16/17)




Solution? It's in the picture! Click on it to Enlarge!

Write that System! 500

Write a System ­


You pay $24.50 for 10 gallons of gasoline and 1 quart of oil at a gas station. Your friend pays $22 for 8 gallons of the same gasoline and 2 quarts of the same oil. Find the cost of 1 quart of oil.


In this case, the equation should be:

10x+y=24.50

8x+2y=22


As the equation.



  1. First, to get the equation, I saw that 10 gallons of gas and 1 quart of oil would be the 2 x and y variables because they are the 2 things that add up to the price of $24.50. So I wrote it like this, 10x+y=24.50
  2. I then saw that 8 gallons of gas and 2 quarts of oil was what caused the price or what you would say to be the Independent variables of the equation. So then I wrote, 8x+2y=22.
  3. I had now solved the requirement to turn this word problem into 2 equations.



  1. To find the cost of 1 quart of oil, you would have to first figure out what method you would like to use to solve this equation. I used elimination because I only wanted to solve for the cost of one quart or for y.
  2. So what I did was to use the method of elimination. I chose to eliminate 10x and 8x because I only need to solve for y. So I found the LCM of 10 and 8 and got 40.
  3. After that, I multiplied the 2 equations by the numbers that were required to turn 10x and 8x into 40x. Example: (10x+y=24.5)(4) and (8x+2y=22)(5)
  4. Saying that, I then got the equations: 40x+4y=98 and 40x+10y=110.
  5. Since I wanted to get rid of the X variable, I had no choice but to multiply one of the equations by -1 so that I could cancel the 40x out. It looks like this (-1)(40x+10y=110)
  6. Then I added the 2 equations and got 6y=12
  7. I divided 6 on both sides to cancel it out.
  8. y=2 or the cost per quart is $2




Solution? It's in the picture! Click on it to Enlarge!

Word that problem! 500

Word Problems! ­


A rental company charges a flat fee of x dollars for a floor sander rental plus y dollars per hour of the rental. One customer rents a floor sander for 4 hours and pays $63. Another customer rents a floor sander for 6 hours and pays $87. Find the flat fee and the cost per hour for the rental.

Answer to Word that problem you say?

Here, I had to make an equation first before solving and finding the flat fee and cost per hour for the rental, or in this case, the x and y variables.


  1. The flat fee represented x dollars and the dollars per hour, represented y dollars.
  2. So I found the number of hours rented which is 4 and the total cost is $63 dollars: the equation is then, x+4y=63.
  3. I did the same with the other equation and got: x+6y=87.
  4. I decided to used the elimination method since I thought it would be the most appropriate thing to use at that time.
  5. I chose the variables that I wanted to eliminate to be x and x or 1x and 1x, their LCM is 1.
  6. The equations are still the same, x+4y=63 and x+6y=87
  7. Since I wanted to eliminate, x by canceling it out, I multiplied x+6y=87 by -1. Example: (x+6y=87)(-1)
  8. After that I added the 2 equations and got -2y=-24
  9. I divided -2 on both sides and got:
  10. y=12


  1. To find x, I already got y so I just plugged in the y
  2. x+4y=63 ---> x+4(12)=63
  3. 4 times 12 equals 48 so: x+48=63
  4. I subtracted 48 from both sides and got:
  5. x=15


The solution is in the picture on the right! Click on it to Enlarge!

Extra things to know!!!!

When should I use substitution?

It is best to use substitution when you have 2 equations, one that is in slope intercept form and the other in standard format. That way all you will have to do is plug in the numbers! For Example:


2x+3y=10

y=x+4


So now all I would have to do is replace the y in the 2x+3y=10 with x+4 since y= x+4.


It should look something like this:


2x+3(x+4)=10


And then you solve it!

When Should I use elimination?

Best be for you, if you use elimination when both equations are in standard form or are easy to turn into standard form such as you can turn y=3x+2 into -3x+y=2 easily in a few simple steps. After that all you have to do is find the least common multiples of the 2 numbers that you want to eliminate. Also you can use this when you want only one answer such as what x equals or y equals.


For Example:


2x+3y=12

2x+1y=8


Since both equations have 2x, I would choose 2x from the two equations as the LCM of the 2 and then I would simply continue the process of solving the equation. (Look at the Elimination picture at the beginning of the page)


Another Example:


3x+2y=4

y=4x+2


Then you would simply convert y=4x+2 into standard format which would be -4x+y=2. After that, you would solve the equation by eliminating one of the variables that you want by finding the LCM of 2 of the same variables in the equations.

When Should I use Graphing

Graphing in the system of equations is considered pretty easy of when you use it to solve equations but first of all when should I use it?


  1. No brainer, when you have a CALCULATOR or a GRAPHING DEVICE of some sort.

Basically after that, another time when you should use it is when you have the 2 equations in slope intercept format (LUCKY YOU) or if you are able to turn the 2 equations easily into slope intercept format. Another time when it is recommended that you use Graphing to solve the 2 equations is that if you want a quick simple method that can give you the answer after you reach the requirements, once again saying this, of turning the equations into slope intercept format if you haven't already or if the equations are already in that form.


Example:


2x+5y=16

10y=6x+12


In this equation, it isn't really recommended that you use graphing in this equation because it would probably take a while for you to convert both equations to slope intercept format (SIMPLIFIED) before you can insert the data into the calculator and let it graph the lines/equations for you.


Another Example:


y=2x+4

y=6x+12


This would be highly recommended that you use graphing in this situation since both equation are already in simplest slope intercept format. That means all you have to do is to plug in the 2 equations, hit graph, find the intersection point, and VUALLA! There you should find the (x,y) coordinates of where the lines intersect and there's you answer.