Solve Equations with Substitution

Substitution is the process of replacing a variable in an expression with its actual value. If you are given an equation like \(4z + 6 = x + z\), told that \(z = 2\), and asked to solve for x, what do you do? The first step is to substitute 2 for every z in the problem:

$$ 4z+6=x+z $$ $$ \cancel{4z}4*2+6=x+\cancel{z}2 $$

This leaves us a more manageable equation, \(4 * 2 + 6 = x + 2\). After simplifying things a little we get \(14 = x + 2\). Subtract 2 from each side and you'll discover that \(x = 12\).

Substitution can also involve more complicated problems, like \(5x^2 + 2y -6 = z\), where \(y = -1\) and \(x = 4\). Just substitute in the same way, placing the value of each variable in place of the letter and simplifying.

$$ 5x^2+2y-6=z $$ $$ 5*4^2+2(-1)-6=z $$ $$ 5*16-2-6=z $$ $$ 72=z $$

Those example are pretty straightforward. You not even have needed that explanation, but it leads to more complex examples. The idea was first to understand the concept of substitution before we go further. There is another form of substitution where we insert a variable to stand for a particular expression. In some cases you will be dealing with incredibly complex problems, and it may be easier to substitute a variable, like y for part of the problem. Then, when you have solved the value of y, you can plug back in the original expression.