Unit 2 · Lesson 2.4

Variables on both sides

Some questions put the unknown on both sides at once. "When does plan A cost the same as plan B?" puts x in both prices. You already know almost all the moves for this. There's just one new first step: get all the x's onto one side. After that, it's the same kind of equation you've already solved.

Picture the balance scale again, but now both pans hold boxes and coins. Take 5x + 2 = 3x + 10: the left pan has five boxes and 2 coins, the right has three boxes and 10 coins. Your goal is to get all the boxes onto one pan. Do it by taking the same thing off both pans. Take 3x off each side. Now the right side has no boxes left. You're left with 2x + 2 = 10, a two-step equation you already know how to solve: $$\begin{aligned} &5x+2=3x+10 \\ &\xrightarrow{\,-3x\,}\; 2x+2=10 \\ &\xrightarrow{\,-2\,}\; 2x=8 \\ &\xrightarrow{\,\div 2\,}\; x=4 \end{aligned}$$ $$\text{Check: } 5(4)+2=22 \;\text{ and }\; 3(4)+10=22$$ A tip: move the smaller x-term first (here the 3x, not the 5x). That keeps the number in front of x positive, which is easier to work with. Moving the other one also works. You get the same answer either way.

The three outcomes

So far every equation has had exactly one solution. That's the common case, but it isn't the only one. Now that x is on both sides, the same steps can lead to three different results. You can tell which one it is by watching what happens to x as you gather terms:

• Conditional: exactly one solution. The x's don't cancel out. You get x by itself and one value for it. This is what you've been doing all along.
• Identity: infinitely many solutions (all real numbers). The x's cancel out on both sides, and what's left is true (like 4=4). That means the two sides were equal the whole time. So every number works.
• Contradiction: no solution. The x's cancel out, and what's left is false (like 3=5). Nothing you plug in can make 3 equal 5, so there's no answer.

You don't need to guess the answer type ahead of time. Just solve as usual. If the x's disappear, stop. Then ask: is what's left true or false? Think of the scale. If both pans hold the exact same thing, it stays balanced for any value of x. That's an identity. If the pans can never match, nothing balances them. That's a contradiction.

Here's one worked example of each outcome. Keep the substitution habit even when the variable disappears.

Two valid solving orders (which is cleaner, and when)

Often there's more than one good way to start. Both ways give the right answer, so you can't go wrong. Take 3(x+2)=18. Both of these are right:

2.4.w8 $$\begin{aligned} &\textbf{Distribute first:}\quad 3(x+2)=18 \\ &\xrightarrow{\text{distribute}} 3x+6=18 \\ &\xrightarrow{-6} 3x=12 \\ &\xrightarrow{\div 3} x=4 \end{aligned}$$

$$\begin{aligned} &\textbf{Divide first:}\quad 3(x+2)=18 \\ &\xrightarrow{\div 3} x+2=6 \\ &\xrightarrow{-2} x=4 \end{aligned}$$

Same answer (x = 4, and 3(4 + 2) = 18 checks). 3 goes into 18 evenly (18 / 3 = 6). When the number out front divides the other side evenly like that, divide first. It takes fewer steps and keeps the numbers small. But sometimes the number doesn't divide evenly. Take 3(x + 2) = 20. If you divide first, you get x + 2 = 20/3, and that fraction follows you through every step. In that case, distribute first to avoid fractions. With practice you'll start to spot which way is quicker. Until then, either one is fine.

2.4.w9 One spot-the-error to read. Below is a solution to 5 − 2(x − 1) = 9. One line in it is wrong. Watch what happens when you distribute a negative number. Find the line where it goes wrong: $$\begin{aligned} &5-2(x-1)=9 \\ &\to\; 5-2x-2=9 \\ &\to\; 3-2x=9 \\ &\to\; -2x=6 \\ &\to\; x=-3 \end{aligned}$$ The break is in the very first move. The −2 multiplies the −1 inside. −2 × −1 = +2, not −2. So the correct second line is 5 − 2x + 2 = 9. That becomes 7 − 2x = 9. Then −2x = 2. So x = −1. If you didn't spot it, that's fine. The fix is right here. Confirm x = −1 in the original: 5 − 2(−1 − 1) = 5 − 2(−2) = 5 + 4 = 9, which matches.

Two common mistakes to watch for. When you subtract an x-term, take it off both pans. Taking 3x off only the left side is the most common mistake here.

Watch the sign when you multiply into a subtraction. In 2(x − 1), the answer is 2x − 2. It's not 2x − 1 or 2x + 2. The 2 times the −1 gives −2. If you're not sure, check your answer by plugging it back in.

If this lesson feels like a lot, that's okay, it really is a lot. It uses everything from the earlier lessons at once. It's fine to leave a problem and come back to it tomorrow. A break really does help, and you haven't lost anything. When you come back, re-read the balance-scale part at the top first. Remember: every move is still just taking the same thing off both pans.