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Solving Systems of Equations using Substitution - Concept

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

A system of equations is two or more equations that contain the same variables. Solving systems of equations by substitution is one method to find the point that is a solution to both (or all) original equations. Besides solving systems of equations by substitution, other methods of finding the solution to systems of equations include graphing, elimination and matrices.

Anytime you're asked to solve a system of equations keep in mind you have a few options for how to do the problem. One option is by graphing another option is by substitution which is what we're going to look at now. You might try elimination or matrices.
Let's talk more about substitution, there's two different ways to go about it. One way to do substitution is to put both equations into y=mx+b form and then you write these expressions as equal to each other in the equation. It's kind of like this idea, you guys know that in the United States 4 quarters is equal to one dollar right? 4 quarters is a dollar, I could also say 10 dimes is equal to a dollar. So what this process is doing, using substitution you're going to write something along the lines of 4 quarters is equal to 10 dimes. I use d and q to abbreviate but I think you guys get the idea, it's the substitution piece where these two expressions become equal to each other if they're both equal to y, that's one option to do, for how to do substitution.
The other way to do substitution is to solve for one variable and then substitute the expression into the other equation. Like for example if I have something like y=2x-3 and 4x-2y=4 something like that I just made that up. And I wanted to solve using substitution; I know that y could be replaced with 2x take away 3. So instead of y right there I'm going to write the whole quantity 2x-3 and then solve the equation. I'm not going to do this problem all the way I'm just want to set it up so you guys can see how substitution works. You substitute this expression into where that equation has the letter y. And now I have one equation with one variable it only has x, and that's something I know how to solve.
Either way no matter how you chose to solve this system of equations with substitutions, please make sure your answer is a point. Meaning it has an x value and a y value, the solution to a system of equations is the point where the lines cross. You need to find not only half like the x or the y, you need to find both. Your final answer will always look like some x number comma some y number.
So again you always have a choice for how to solve systems of equations, substitution is effective if you have either one letter already solved, meaning it's all by itself or if you have two equations that are in y=mx+b form.

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