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Systems of Equations

Tutorial 1

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Elimination method
Substitution method
Example 1
Example 2


A system of equations is defined as 2 or more equations considered together. One such example is below:

part 1

The bracket at the left is the notation to state that these equations are together and are considered a system of equations.

Elimination Method

There are a few ways that you can solve a systems of equations. The most used method is the elimination method. This states that if we know how to solve for one of the variables in our system, we can eliminate it and solve for the remaining variable.

Let's start with the system above as our first example. Which variable seems easier to eliminate? If you said the y variable, you are correct. The way to eliminate that variable is to add the 2 equations as so:

   4x + y = 13
 + 5x - y = 14

We would get the following general equation:

9x = 27

and we can conclude that the x value is 3. Going back to the original equations, we get the following:

part 2

without doing any extra work, we can easily see that y is equal to 1.

Our last step of this method is to check the equations and to make sure each of them are equal to their respective values:

Check for equation 1:

4x + y = 13
4(3) + (1) = 13
12 + 1 = 13
13 = 13

Check for equation 2:

5x - y = 14
5(3) - (1) = 14
15 - 1 = 14
14 = 14

Here is a general outline for the elimination method:

1. Put the equations in standard form (eliminating 
   fractions and getting the variables to one side).
2. Decide which variable you want to eliminate.
3. Multiply one or both equations so that by 
   appropriate constants so that you are dealing with 
   opposite coefficients.
4. Add the resulting equations.
5. Solve the equation for that one variable.
6. Plug in that value from step 5 to any of the 
   equations to get the last value.
7. Check your result.

Example 1:
Solve the following system

example 1a

Here, we need to put these equations in standard form. What that means is the variables are on one side while the values are on the other. After some manipulations, we get the following:

example 1b

In order for you to eliminate a variable, the coefficients must cancel out to be 0. As of now, there is no way for that to happen. However, if we choose to eliminate the x value first, we would see that 6 would need to be the coefficient (6 is the LCM of 3 and 2). Therefore, we multiply the top equation by 2 and the bottom by 3 to get the coefficients equal to 0.

example 1c

And now we add the equations on the right above to get:

7y = 0

And we can clearly see that y is 0. Now go back to the original equations and plug in:

example 1d

And we can now see that the x value is 2. But does this check?

Check for equation 1:

3x - 4y = 6
3(2) - 4(0) = 6
6 - 0 = 6
6 = 6

Check for equation 2:

-2x + 5y = -4
-2(2) + 5(0) = -4
-4 + 0 = -4
-4 = -4

Thus, the solution is (2, 0) representing (x, y) respectively.

Substitution Method

When you are given a systems of equations where one variable is given an expression, you can use this substitution method. Let's say you are given the following system:

part 3

We can see that we are given an explicit value of x which is 3y - 4. We can now use that value and plug into the second equation to find the value of y:

3(3y - 4) + 2y = 10
9y - 12 + 2y = 10
11y - 12 = 10
11y = 22
y = 2

And we can see that y = 2. Now let's go back to the original and plug in to get the following:

x = 3(2) - 4
x = 2

We can now see that x is also equal to 2. Let's also go and check as usual:

Check for equation 1:

x = 3y - 4
(2) = 3(2) - 4
(2) = 6 - 4
2 = 2

Check for equation 2:

3x + 2y = 10
3(2) + 2(2) = 10
6 + 4 = 10
10 = 10

Thus, the solution is (2, 2)

Example 2:
Solve the following system.

example 2a

Here, we will use the first equation to get an explicit value of y since there is a coefficient of 1. We can still use the bottom equation but this one is easier to work with.

example 2b

Now take this value and plug it into the second equation and solve:

example 2c

HUH?!? Is this even possible? The answer is YES! There may be systems of equations where there will not be values that produce the answer as in the above. This is called an inconsistent system.

To see why, if you were to solve the second equation for y, here is what you would get:

example 2d

Comparing that to the original, we notice a difference in the y-intercept value. That means the lines in this system do NOT meet on a graph and thus will produce two separate values.


After viewing this tutorial, you should be able to solve the following challenges:

Algebra 11
Algebra 12
Algebra 13
Algebra 14

Good luck! Remember to register for a FREE account today to submit the answers to the challenges!


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