# Solving Equations with the Substitution Method: Algebra Examples & Overview

Instructor: Yuanxin (Amy) Yang Alcocer

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.

## The Substitution Method

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?

## How to Use It

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.

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