# A Complete Guide for Solving Systems of Equations by Substitution

Get the free Solving Systems of Equations by Substitution worksheet and other resources for teaching & understanding Solving Systems of Equations by Substitution

**Here’s How to Solve Systems of Equations by Substitution**

**Solving Systems of Equations by Substitution** is a method to solve a system of two linear equations. **Solving Systems of Equations by Substitution **follows a specific process in order to simplify the solutions. The first thing you must do when **Solving Systems of Equations by Substitution** is to solve one equation for either variable. Next, you must take the solution for the variable and substitute it into the other equation for the variable. Then you solve for the variable and get a numerical solution. Use the solution of the variable to substitute back in to either original equation. The last step when finding out how to solve** Systems of Equations by Substitution** is to solve for the last variable by following order of operations.

**Common Core Standard: **8.EE.C

**Related Topics:**Identifying One, None, Infinite Solutions, Solving Systems by Graphing, Solving Systems by Elimination

**A Quick Substitution Method Definition**

Here’s the quick definition of how to solve system of equations by substitution. Solving Systems of Equations by Substitution is a strategy to explain an set of two equations. The first thing you should do when Solving System of Equations by Substitution is to solve one mathematical statement for either variable. Next, you should take the answer for the variable and substitute it into the second mathematical statement for the variable. At that point you simplify for the variable and get a numerical answer. Use the simplified variable to substitute back in to either the first or second mathematical statement. The last part when using Substitution Systems of Equations is to simplify for the last variable by following the rules for order of operations.

**5 Quick steps to Answer any Substitution Method Example Problem**

- Solve one equation for the x or y variable.
- Substitution that solution into the other equation for the variable that you solved for (either x or y)
- Solve the new equation for the variable to get a numerical answer.
- Take the numerical answer and substitute it back into either equation to solve for the remaining variable.
- Put your answers into an x and y coordinate.

**Solving Systems of Equations by Substitution Practice Problems Quiz**

**Watch the video where we complete our Systems of Equations by Substitution Worksheet**

Watch our free video on how to solve **Systems of Equations by Substitution**. This video shows how to solve problems that are on our free **Solving Systems of Equations by Substitution **worksheet that you can get by submitting your email above.

**Watch the free Solving Systems of Equations by Substitution video on YouTube here: How to Solve Systems of Equations by Substitution**

**Video Transcript:**

This video is about solving systems of equations substitution. You can get the solving systems by substitution worksheet used in this video for free by clicking on our link in the description below.

Here we are at a first problem. Anytime you’re trying to solve a system of equations your answer must be in the form of an X and a y-coordinate. That means anytime you solve this by any method whether it’s graphing elimination or substitution you have to have a coordinate as your answer. When solving systems like this you will solve by substitution using the substitution method algebra. That means you already have to know how to solve by substitution.

Here we learn our first problem on our systems of equations substitution worksheet. We have 3x minus y equals 7 and then y equals 2x plus 2. Now when you’re solving systems by substitution you have to use substitution to substitute one equation in to the other equation. In the case of this first problem, our second equation here is already solved for the variable Y. We already know Y is equal to 2x plus 2 because this is already solved for y. We can take 2x + 2 and we can substitute it in for Y in the other equation. What we’re going to do is we’re going to take our top equation and we’re going to rewrite it with our second equation substitute it into it. We have our second equation which is 2x + 2 and we know Y is equal to that.

We substituted it in for where Y used to be now we have to solve for X. That means we have to simplify this. We’re going to go ahead and distribute this negative to everything inside the parenthesis. Now we have 3x minus 2x minus 2 because we do negative 1 times 2x and then negative 1 times 2 equals 7. Now the next step is to combine like terms. We do 3x minus 2x and 3x minus 2x is 1/x. You bring down your -2 and your 7 and then we’re going to add 2 to both sides so that we cancel this too and we end up with 1x or just x equals 9.

Now we know that X is equal to 9 and you remember you have to have an X and a y-coordinate. We take our X is equal to 9 and we’re going to substitute that back in to X in our bottom equation. We’ll take y equals 2 times our x value which is 9 plus 2 y equals 2 times 9 which is 18 plus 2. Y equals 18 plus 2 which is 20. Now our complete solution is x equals 9. Our x coordinate is 9 and then our y coordinate is 20. We’ll use X is 9 y is 20 and that is the coordinate that is our solution.

The next problem we’re going to go over on our system on our solving systems of equations by substitution worksheet is number 3. Now this time we have negative x plus 3y equals 3 and then the second equation is X minus 2y equals 8. Now this time we don’t have any variable that has already been solved for. What we have to do is we have to pick a variable to solve for so that once it’s solved for we can substitute it into the other equation. You can solve for any variable, it’s up to you. You could solve for this one you could solve for this Y, you could solve for this X or you could solve for this 2y. Easiest thing to solve for would be to solve for this X down here because all you have to do is add 2y to both sides and these will cancel and you solve for x automatically.

Now we know X is equal to these two terms if we tried to solve for any other variables we would have had a coefficient here and we would have had to divide by everything by that coefficient. This was just the easiest variable to solve for. We know x is equal to 2y plus 8 we have to take this 2y plus 8 and we have to substitute it to our second equation or top equation in this case. When we do that we’re going to take our equation which is negative x or in this case negative 2y plus 8 because that’s what we’re substituting in for and then plus 3y. We bring the rest over. Now we know it’s equal to negative 2y plus 8. We take this negative and we distribute it to everything on the inside of the parenthesis. This will be negative 2y negative 1 times 8 minus 8 plus 3y equals 3. The next step is to combine like terms so we’re going to add these two together negative 2y plus 3y is 1y. You bring down your minus 8 and then you bring down your equals 3. Then we have to add 8 to both sides you get Y. 1y or just y equals 11.

We know Y is equal to 11 so now we’re going to take y equals 11 and we’re going to substitute 11 in for the Y back in the equation we simplified for X. We have x equals 2 times we know Y is 11 so we’re going to substitute 11 in for y plus 8 2 times 11 is 22 Plus 8. And then when you solve 22 Plus 8 you will get 30. The solution because we have to have a coordinate as x equals 30 y is 11 and that’s our total solution. You can try these practice problems by downloading the free solving system of equations by substitution worksheet above.

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