Systems of Equations

In this section we'll talk about systems of equations. What is a system? This is when we have multiple unknowns and multiple equations, and we want to solve for all unknowns.

Systems

Recall in the previous section, we solved equations with 1 variable. What if we had 2 variables and 2 equations? For example, suppose we know that y=3x-6, and y=2x+8. How do we solve this?

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Idea:

What does this even mean? Well, we have 2 variables, x and y. We can notice that both of our equations are linear equations. We want to find a coordinate (x,y) that satisfies both equations. In other words, we're trying to find the intersection of the 2 lines.

Example:

We'll continue with the first example. We know y =3x-6 and y = 2x+8. We notice that y is isolated in both equations. This means 3x-6=2x+8 ! From here, we'll use the skills we learned in the previous section to get that x=14.

Are we done here? No: we solved for x, now we need to solve for y. Well, we can simply plug x back into any of the equations. For example, y=3x-6, so y = 3(14)-6 = 36.

This tells . This example was a specific case of using the substitution method, which we'll go into soon.

Graphical Example

Find the Solution to the System of equations with the following equations.

1.  13x-29 = 2y-42

2. 7x-4 = 2y - 16

What we can do here is graph both equations, and simply look for the intersection! Since they are both linear equations, it's not that hard to graph and find their intersection. Take a look at this illustration.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Based on the diagram, what's the solution to the system? Be exact!

Substitution

Now, we'll talk about substitution, one of the most common methods for solving systems of equations. Substitution works by isolating x in terms of y (or vise versa), and plugging this expression into another equation. Let's see it in action.

Example 1:

 

Solve the system 2y - x = 7, x = 4y +9                                      

Step 1: Isolate a variable. We already have x isolated in the second equation. 

Step 2: Plug in expression to the first equation. We already know what x is in terms of y. So, we'll substitute this value into the first equation like so:  2y - (4y+9)=7.

Step 3: Solve the new equation for 1 variable. Since our new equation only has y, we can easily solve and find that y=-8.

Step 4: Solve for the other variable! We have that y=-8. So we'll plug this value into the first equation to see that

-16-x=7, meaning x = -23

Concluding:

We now know that the solution to the system is (-23,-8)

Elimination

Now we'll talk about elimination, another common method for solving these systems. Using elimination involves adding or subtracting the 2 equations, in order to eliminate one variable. Let's see it in action, using the same example.

Example 1:

Solve the system 2y - x = 7, x = 4y +9 

 

Step 1: Find the variable to eliminate. In this case, we see that in both equations, x has a 1 and -1 coefficient, so it should be easy to eliminate.

Step 2: In this step, we'll actually perform elimination. We add both equations together. On the left hand side, we get 2y-x+x, and on the left hand side, we get 7+4y+9. Remember that these sides should be equal. So we actually have 

2y-x+x = 7+4y+9. 

Step 3: Solve! This is the beauty of Elimination. Adding the 2 equations, we see that the x's cancel. So we are left with 2y = 4y+16, and y=-8. 

Step 4: This is the same step as in Substitution, we'll simply plug in our known value to get the value of x. We get that x=-23

Concluding:

We now know that the solution to the system is (-23,-8)

You try it!

1. Use elimination to solve the system x-7y=-11, 5x+2y=-18

2. Use substitution to solve the system x-7y=-11, 5x+2y=-18

3. Use elimination to solve the system 7x-8y=-12, -4x+2y=3

4. Use substitution to solve the system 7x-8y=-12, -4x+2y=3

5. Discuss the advantages and disadvantages of both methods, especially with regard to these 2 systems.