# Unit #5: Systems of Equations

This unit focuses on solving systems of linear equations. A system is a set of two or more equations. In our case, these two equations will be linear. (Hint: Linear means they form a line when graphed!) We will be discussing four different ways to solve systems of linear equations. From now on I will be using the abbreviation, SOE, to represent systems of equations.

We will continue to use the virtual nerd website to find short tutorials to help learn these concepts. I have also included some additional websites that have some completed example problems, as well as a link to iXL website for practice problems for each section. The button below will take you to the website and you can browse around, OR you can use the specific links below to get started.

# Solving SOEs by Graphing

As the title of this section suggests, we will be graphing these linear equations to find solutions. The button below will bring a FREE online graphing calculator that you can use. Review the information below to find out more about solving SOEs by Graphing.

**Check 4 Understanding (C4U)**: use the link below to access some online practice problems that address the concept topic.

# Solving SOEs by Substitution

You have all experienced substituting numbers into equations, but now we are taking it a step further. We are substituting whole expressions into equations to help us determine the solution for the SOE. Review the information below to learn more about this method.

**Check 4 Understanding (C4U)**: use the link below to access some online practice problems that address the concept topic.

# Checking your Solution

Now that we have covered two different ways to solve SOEs, it is important to know how to check if your answer is correct. The information below will help you determine whether your answer is the correct solution to the SOE.

# Solving SOEs by Elimination

In this method of solving SOEs, we will be eliminating a whole variable from the system. First we will look at a simple elimination method using Addition and Subtraction. Next we will look a more complex method of elimination using Multiplication. Remember, once you find the value of one variable, you will still need to find the value of the other variable. You can substitute the variable that you do have into either one of the equations you started with. Then solve for the missing variable. Always write your solutions as a point (x, y).

# Addition Method

If the coefficients are the same number but opposite (one positive and one negative), you will add the two equations.

# Subtraction Method

If the coefficients are the same, then you will subtract the two equations.

The video below provides some great examples with block visualization of the addition and subtraction methods for elimination.

The link below shows examples of the addition method, the subtraction method, and the multiplication method (which will be covered tomorrow).

# Multiplication Method

In order to use the multiplication method, you will need to have a SOE where no coefficients are the same in either equation. Thus, we will need to multiply one or both of the equation by a number in order to force one set of coefficients to match (either the x or the y coefficients.)

The video below will help you understand how to solve a system of equations using elimination and multiplication. Notice how the coefficients are forced into matching.

**Check 4 Understanding (C4U)**: use the link below to access some online practice problems that address the concept topic.

# Three Ways to Solve SOEs

So far we have covered three different methods for solving SOEs: graphing, substitution, and elimination. The video below will cover all of these different methods in one place!

# Systems Word Problems

Now that we have covered the three different methods for solving SOEs (graphing, substitution, and elimination), let's apply those techniques towards solving word problems where two different variables exist. There are three steps that will help you stay organized throughout this process.

**Step #1 -- DEFINE VARIABLES!**

This step is important so that we know exactly what we are going to be solving for and determine reasonableness of the solutions that we receive.

**Step #2 -- WRITE SYSTEM OF EQUATIONS & SOLVE!**

This step will require you to write two equations using the variables that you defined in the previous step. As far as identifying which method to use for solving the SOEs, you have that choice to make. Some instances will be easier/quicker to solve by elimination, some will be easier/quicker to solve by substitution. You make that determination.

**Step #3 -- IDENTIFY THE ANSWERS!**

This last step is important. Instead of just saying that a certain variable equals a particular value, you will need to say exactly what that variable represents. This will be a short statement or a few sentences that relate the solution back to the problem itself.

VIDEO - SOLVING WORD PROBLEMS WITH TWO EQUATIONS

VIDEO - SOLVING WORD PROBLEMS USING ELIMINATION

**Check 4 Understanding (C4U)**: use the link below to access some online practice problems that address the concept topic. The first set of problems is dedicated towards Solving SOE Word Problems using Substitution. The second set of problems are dedicated towards Solving SOE Word Problems using Elimination. The last set of problems are dedicated towards Solving SOE Word Problems using Any Methods.

# Linear Inequalities

Linear inequalities are the same as linear equations, except they have inequality symbols in place of the equal sign. Solutions to linear inequalities are all ordered pairs that make that inequality statement true.

There will be times that you will be asked if a specific point is part of the solution set for a linear inequality. Watch the video below to gain a better understanding of this process.

**Check 4 Understanding (C4U)**: use the link below to access some online practice problems that address the concept topic.