What is the Substitution Method?
The Substitution Method
In this video, I want to look at the substitution method for solving systems of equations. Let’s take the example of 3x plus 5y, equals 15, and 2x minus 4y, equals 4. We have a system of equations and we’re going to solve it using the substitution method.
Now the first thing you want to do with the substitution method is to pick one of the two equations, and solve it for one of the two variables. Based on the equations that we have here, I’m going to pick the second equation, and I’m going to solve it for x. To do this, the first thing I want to do is divide each term by 2.
That will give us 2x over 2 is x, 4y over 2 is 2y, and 4 over 2 is 2. We can further solve this by adding 2y to both sides to isolate x, and give us x in terms of y. That leaves us with x equals 2y plus 2. Now that we have this equation, solve for x in terms of y, we can now substitute this value, which is equal to x, into the -into the- first equation up here in place of x.
We can rewrite this equation now as 3 times 2y, plus 2, plus 5y, equals fifteen. When we solve this equation for y, that will give us the value of y in both of these equations. Let’s go ahead and distribute the 3, and that will give us 6y plus 6, plus 5y, equals 15. We can combine the two y terms, and that gives us 11y.
We’ll subtract 6 from both sides and that will give us 9 on the right. y is equal to 9 over 11. This is of value y in these two equations. Now that we have a y, we can substitute this value into the equation for y, and solve for x.
Let’s do that, let’s take the second equation again. We have 2x minus 4, times 9/11, is equal to 4. We can go ahead and multiply this out, 2x minus 4, times 9, is 36. 36 over 11 equals, we’ll go ahead and convert this to 11ths, and that will give us 44/11. Now we need to add 36/11 to both sides, and that will isolate the x term.
That will give us 2x equals 80/11, or x equals 40/11. This is the substitution method. You solve one of the two equations for one of the variables, and then you substitute the value of that variable in terms of the other variable into the other equation.
You solve that equation for one of the variables, and plug that value into one of the two equations to solve for the other variable. That’s the substitution method.