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High School Algebra II: Tutoring Solution26 chapters | 274 lessons

Amy has a master's degree in secondary education and has taught math at a public charter high school.

What does the substitution method have to do with substitute teachers? Learn why you should love substitutes. Also learn how substitutes make your life easier when it comes to solving sets of equations.

It shouldn't give you a headache. Really, it shouldn't. The **substitution method** helps you to solve your problems by substituting simpler things into your equation so you can solve the problem faster and without difficulty. Just like real-world substitutes take the place of your teacher and make your life a little bit easier for the day, the same goes for the substitution method. Also, similar to the way substitute teachers give you direction while your teacher is gone, the substitution method gives you a clearer idea of how to solve the problem when the answer is not present.

To give you an idea of how substitution works, let's say that I have some nickels and dimes. I want to figure out how many nickels and dimes it takes to make a quarter. Okay, so you have probably already figured out that it takes two dimes and one nickel. But let's say that I didn't know that. Let's say that the only things I did know was that I had nickels and dimes and that it takes two nickels to make a dime. Instead of trying to figure out how many dimes and nickels I need, I can much more easily try to figure out how many nickels I need if I replace every dime with two nickels. I figure out the number of nickels by dividing 25 by 5. Doing that, I get 5 nickels. But wait, I know that I can replace two nickels with a dime, so how many dimes can I substitute back in? I can switch four of the nickels out and I will have two dimes. And voila, I have my answer. It takes two dimes and one nickel to make a quarter. See how it works?

Substitution is usually used in systems of equations. If you are given a word problem, you first need to write it as a system of equations. Remember our example of the nickels and dimes? How would you write that as a system of equations? I would start by defining my variables. I will label my dimes *d* and my nickels *n*. In this example, I end up with two equations: one that gives me the number of nickels and dimes it takes to make a quarter and another that tells me how many nickels are in a dime. My two equations are *10d+5n=25* and *d=2n*. Notice how I've multiplied the dimes by 10 and the nickels by 5 in the first equation? I did this because I need to know how many pennies are in each group. I know that dimes are 10 cents each and nickels are 5 cents each. So, to figure out how many pennies are in each, I multiply.

Looking at the first equation, I notice that I have two variables. Hmmm - I can't solve that because I have more variables than I know what to do with. But, looking at the second equation, I see that I can substitute *2n* for the *d*. Doing that, my first equation becomes *10(2n)+5n=25*. I can definitely solve this for *n*. Let's see what we get.

10(2n)+5n=25 | I multiply the 10 and 2 |

20n+5n=25 | Combine like terms |

25n=25 | Divide by 25 on both sides |

n=1 | I get one nickel |

This tells me that I need one nickel. To figure out the number of dimes, I plug my *n=1* into the second equation. This is what I get.

d=2n |

d=2(1) |

d=2 |

So I need two dimes. So my final answer is two dimes and one nickel to make a quarter. While the math problems you see may seem harder, they really aren't. Just remember how easy this was. You are doing the exact same steps, only the variables might be different.

If you happen to have three variables instead of two, just do what you just learned. The only major difference is that you need a third equation to tell you what to substitute for the third variable. Remember, the only way to solve an equation is if it only has one variable. So that is your goal - to substitute other equations for the unknown variables so that you are left with only one variable. For example, let's say your math problem had *x*, *y*, and *z* as variables. You see that the second equation gives you *y* in terms of *x* and the third equation gives you *z* in terms of *x*. Well, you could substitute those equations into the first so that the only variable it has is *x*. Once you've solved for the *x*, you can use that information to solve for the other variables.

The substitution method is not as hard as it may first appear to be. Remember that your goal is to simplify your life. You know that to solve an equation, it can only have one variable. Make that your goal when looking at the other equations in the problem. You want to use an equation where you can substitute or rewrite an unknown variable in terms of the variable you want to solve for.

After finishing this lesson, you should be able to:

- Describe the substitution method
- Explain how to use substitution to solve equations
- Solve examples by using the substitution method

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High School Algebra II: Tutoring Solution26 chapters | 274 lessons

- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- How to Solve a System of Linear Equations in Two Variables 4:43
- How to Solve a Linear System in Three Variables With a Solution 5:01
- How to Solve a Linear System in Three Variables With No or Infinite Solutions 6:04
- Gaussian Elimination: Method & Examples
- Solving Equations with the Substitution Method: Algebra Examples & Overview
- Go to Algebra II - Systems of Linear Equations: Tutoring Solution

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