Algebra Continued

OK. I think we are ready for more complex equations to solve. We have covered the distributive property, fractions, and signs. If you do not understand any of the steps below, please see previous blogs in the “Algebra” category or use the tags to go to any specific area.

Let me introduce another symbol frequently used when doing any maths development. “⇒” means “it follows that” or “after doing some math, you get”. I will use it below.

Let’s solve for x in the equation 3x + 18 = 117. When I say “solve for x” I mean find the value of x that makes the equation a true equation. Using algebra’s prime directive: you can do any legal math to any side of an equation as long as you do the same math on both sides. So we will use this to get the equation to look like x = some number. So what guides you in this process is to do the math on both sides that gets x by itself.

Well the first thing you may want to do to our equation is to get rid of the 18 on the left side. I can do this by subtracting 18 from the left side, but I must do the same on the right:

3x + 18 = 117 ⇒ 3x + 18 -18 = 117 -18 ⇒ 3x + 0 = 99 ⇒ 3x = 99

Did you follow all that? Subtracting 18 from 18 is zero, and zero added to anything is the same anything so the zero can be removed. So what do we do with 3x = 99 to get x by itself? This is where the post on fractions comes in. If I divide the left side by 3 by using fraction notation, I will have a common factor of 3 in the numerator and denominator, and these can be cancelled, that is replaced with 1’s which can be assumed to be there:

$\begin{array}{c} {{3}{x}\hspace{0.33em}{=}\hspace{0.33em}{99}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\frac{3x}{3}\hspace{0.33em}{=}\hspace{0.33em}\frac{99}{3}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\frac{\rlap{/}{3}x}{\rlap{/}{3}}\hspace{0.33em}{=}\hspace{0.33em}{33}}\\ {{x}\hspace{0.33em}{=}\hspace{0.33em}{33}} \end{array}$

So we have come up with the answer that x = 33. Did we make any mistakes? Well let’s check our answer by substituting 33 in place of x in the original equation and see if we get a true equation:

3(33) + 18 = 117 ⇒ 99 + 18 = 117 ⇒ 117 = 117

Well that’s as true as it gets. Good job!