Try this one on your own. Don’t move on until you’re stuck for at least a couple minutes.
Example 2: Solve for \( x \):
\[ 2 + 5(1+x)-3x=7x+8(-2)+13 \]
This looks a little disgusting, right?
Don’t worry! Remember our end goal is to simplify the equation to \( x = ? \). Thus, we should get the variables onto one side and the constants to the other. Let’s do this!
A good first step is to try to simplify both sides and “combine like terms”. That is, we should combine all x’s and constants. Warning: you cannot combine constants and like terms with addition and subtraction! Something like \( 3x+5=8x \) isn’t valid as a step.
We notice the second term on our left-hand side, \( 5 (1+x) \) can be safely simplified into \( 5+5x \) by the distributive property (if this sounds unfamiliar, a look here). Hence, our equation is now
\[ 2+5+5x-3x=7x+8(-2)+13 \]
Let’s combine like terms on the left hand side. We combine the constants( \( 2+5=7 \)) and the variables ( \(5x-3x=2x \)). Also on the right-hand side, we can’t combine variables, but we can do some arithmetic with our constants \( 8(-2)+13=-3 \) (Why?). This is what our equation becomes (the parentheses are inserted to show our simplifications):
\[ (2+5)+(5x-3x)=7x+(8(-2)+13) \]
\[ 7 + 2x=7x-3 \]
Now… both sides are fully simplified (i.e no more like terms can be combined). All we need to do is to move variables onto one side and the constants onto another. We usually do so by eliminating a constant or a variable on one of the sides to leave constants on the other. It doesn’t matter how we do this, as long as we do stuff to both sides. Thus, let’s eliminate the \( 7 \) on the left-hand side by subtracting seven on both sides:
\[ 7+2x-7=7x-3-7 \]
\[ 2x=7x-10 \]
Since only the right-hand side has a constant ( \(-10\) ), we can try to get all the variables onto the left-hand side. A lingering \( 7x \) on the right tells us to subtract \( 7x \) from both sides:
\[ 2x-7x=7x-10-7x \]
\[ -5x=-10 \]
The variables are all on the left, and the constants on the right. YES! We can divide by \( -5 \) on both sides to finish:
\[ \frac{-5x}{-5}=\frac{-10}{-5}\]
\[x=2 \]
Thus, the value of \( x \) that satisfies this equation is \( 2 \).
Try a few more on your own. I encourage you to try the last few. The answers are on the very last page of this post.
- \( 2x + 5 = 15 \)
- \( 3(2x – 1) = 9 \)
- \( 2(3x + 4) – 7 = x + 8 \)
- \( 4x – 5 = 2x + 9 \)
- \( 3(1-x) – 2x = 4(3+5x) + 6 \)
- \( 5x + 3 – 2(x – 1) = 4x + 8 \)
- \( 2(4x – 1) + 3(x + 5) = 6x + 17 \)