An equation is a balance: whatever you do to one side must be done to the other. To isolate x, undo the operations in the opposite order they were applied: addition and subtraction come off before multiplication and division. Then substitute the answer back.
Step 1: Picture It
Start with something you already know.
Socks and shoes
- In the morning you put on your socks.
- Then you put your shoes on over them.
- At night you undo it all.
- The shoes come off first, and then the socks.
- The last thing on is the first thing off.
Solving an equation is the same: undo what was done to x, last thing first.
In 3x + 5, x was multiplied by 3 and then 5 was added. So the 5 comes off first, then the 3.
Where the picture stops: an equation is also a balance, so every move is made to both sides at once.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Rosa thinks of a number. She multiplies it by 4, then adds 3. She gets 31. What was her number?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Clear the brackets first
3(x + 2) = 15: distribute to 3x + 6 = 15, then 3x = 9, x = 3. Or divide both sides by 3 first: x + 2 = 5, x = 3. Both are valid; the second is shorter when the right side divides cleanly.
Variables on both sides
5x + 3 = 2x + 12: subtract 2x from both sides, 3x + 3 = 12, then 3x = 9, x = 3. Collect the variable terms on whichever side keeps the coefficient positive; it avoids a sign error later.
Fractions
x/4 + 3 = 7: subtract 3, x/4 = 4, multiply by 4, x = 16. Or clear the fraction first by multiplying everything by 4: x + 12 = 28. Either way, one operation at a time.
Two special outcomes
2(x + 3) = 2x + 6 becomes 6 = 6: true for every x, infinitely many solutions. 2(x + 3) = 2x + 1 becomes 6 = 1: false, no solution. When the variables vanish, the statement left behind decides.
Justify each step
Write the move beside each line: "subtract 5 from both sides", "divide both sides by 3". The justification is what makes the working a proof that x = 5, not just a guess that happens to work.
Always check
Substitute the answer into the original equation, not into a later line. 3(5) + 5 = 20 holds. It costs one line and catches nearly every arithmetic slip.
Step 3: Try It Yourself
Tap and try it out.
Solve 4(x − 2) = 2x + 6
- 14x − 8 = 2x + 6distribute the 4
Step 0 of 4
3x + 5 = 20
3x + 5 balances 20. Take the same off both pans and they stay level, because the same was done to each side.
Step 4: In Real Life
A phone bill
A bill of 47 dollars on a plan of 20 plus 0.05 a text: 20 + 0.05t = 47. Undo the 20, then the 0.05: 540 texts. Every bill is an equation with one unknown.
Step 5: Watch an Example
One step at a time.
Watch Maya Solve 5(x − 1) = 2x + 13
A bracket on one side and x on both sides. Maya writes the move beside every line.
- Step 1
The 5 multiplies the whole bracket, so distributing it reaches both terms inside: 5x − 5 = 2x + 13. The −5 is 5 times −1, and leaving it as −1 is the most common slip here.
Watch Leo Solve 3(x + 2) = x + 14
The same kind of equation. Leo makes the first two moves, and you finish it.
- Step 1
Distributing the 3 to both terms in the bracket gives 3x + 6 = x + 14.
Two Roads to Solve 4(x − 3) = 24
One equation solved twice: once by distributing, once by dividing first.
- Step 1
Road 1 distributes first. 4(x − 3) becomes 4x − 12, so 4x − 12 = 24. Adding 12 to both sides leaves 4x = 36, and dividing by 4 gives x = 9. Three moves.
Step 6: Your Turn
Practice makes it stick.
The Gym
Problem 1 of 17
A gym charges a $25 sign-up fee plus $15 a month. Mia has paid $130 in all. How many months has she been a member?
months
The Bike Share
Problem 2 of 17
Bike share Plan A costs $10 to join plus $3 a ride. Plan B costs $25 to join plus $2 a ride. After how many rides do the two plans cost the same?
rides
Isolate the Variable
Undo the operations, last one first. Do the same to both sides.
Problem 3 of 17
6x − 4 = 32. What is x?
Problem 4 of 17
2(x − 5) = 14. What is x?
Problem 5 of 17
7x + 2 = 3x + 18. What is x?
Problem 6 of 17
x/5 + 4 = 9. What is x?
Problem 7 of 17
5(x − 2) = 5x − 10. How many numbers make it true?
Whose Step?
One line in each solve is a slip. Find it, then fix it.
Problem 8 of 17
Sam solved 4x + 6 = x + 30. One line is a slip. Which?
Sam's solve of 4x + 6 = x + 30
- 14x + 6 = x + 30
- 23x + 6 = 30− x
- 33x = 36− 6
- Answerx = 12÷ 3
Problem 9 of 17
Solve Sam’s equation, 4x + 6 = x + 30, again. What is x?
Problem 10 of 17
Ana solved 2(x + 5) = 24. One line is a slip. Which?
Ana's solve of 2(x + 5) = 24
- 12(x + 5) = 24
- 22x + 5 = 24
- 32x = 19− 5
- Answerx = 9.5÷ 2
Problem 11 of 17
Solve Ana’s equation, 2(x + 5) = 24, correctly. What is x?
Problem 12 of 17
Dev solved 2(x + 4) = 2x + 5. One line is a slip. Which?
Dev's solve of 2(x + 4) = 2x + 5
- 12(x + 4) = 2x + 5
- 22x + 8 = 2x + 5
- 38 = 5− 2x
- Answerx = 0
Problem 13 of 17
What should Dev have written after 8 = 5?
Pick the Move
Decide the first move before you make it.
Problem 14 of 17
5(x + 3) = 40. Which first move reaches the answer in the fewest moves?
Problem 15 of 17
9x − 4 = 4x + 21. Which move gathers the x terms on one side and keeps the x coefficient positive?
Same Solution?
Decide without solving all the way. Look at the moves.
Problem 16 of 17
Without solving, select every equation that has the same solution as 2x + 3 = 11.
Problem 17 of 17
Select every equation that is true for every number x.
Step 7: Quick Check
Show what you know.
Question 1 of 2
4x + 6 = 26. What is x?
Question 2 of 2
The variables cancel and leave 7 = 7. How many solutions?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
4(x − 1) = 4x + k, where k is a number you choose. For which values of k does the equation have no solution?
Stretch 2 of 2
Find the number a that makes x = 4 the solution of 2x + a = 5x − 9.
What You Learned
- An equation is a balance; do the same to both sides, and write the move beside each line.
- Undo operations in the reverse of the order they were applied; clear brackets first.
- Vanishing variables leave either every solution or none.
- Substitute your answer into the original equation every time.
For the grown-up
The misconception this lesson targets is that solving is a set of tricks done to the numbers you can see, rather than the same move made to the whole of both sides. It shows up three ways: a move made to one side or one term only (the 2 in 2(x + 5) multiplied into the x but not the 5), operations undone in the wrong order (taking 3 from 5(x + 3) = 40 before the 5), and, when the x terms vanish, writing x = 0 instead of reading the statement left behind, which says every number or none. Checking the answer in the original equation, not a later line, catches the first two at once.