One-Step Equations — Print
An equation is a balanced scale, and whatever you do to one side you must do to the other.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Area — Space inside, measured in square units.
A bag of unknown weight balances against seven one-kilogram blocks.
The scale is level, so the two sides weigh the same. That is all "equals" means.
Take two blocks off the right and the scale tips. It was balanced, and now it is not.
Take two off both sides and it stays level. The bag has not changed, but the picture is simpler.
Solving an equation is doing the same thing to both pans until the bag is alone. Every rule in this unit is that one idea.
What the equals sign promises
x + 5 = 12 says that whatever x is, adding five to it gives twelve.
The two sides are the same amount, written two ways.
Solving means finding the number that makes the promise true.
Not an instruction
Equals does not mean "the answer comes next". It means the two sides are equal, which is why you can work on either one.
Undoing with the inverse
To get x alone, undo whatever was done to it, using the opposite operation.
Addition is undone by subtraction; multiplication by division.
x + 5 = 12 becomes x = 12 - 5 = 7.
Both sides, every time
Subtracting five from only one side breaks the balance. The equation stops being true, and so does everything after it.
Multiplication and division
4x = 20 means four lots of x, so divide both sides by four to get x = 5.
x3 = 6 means x shared into three, so multiply both sides by three: x = 18.
The operation in the equation tells you which inverse to use.
Divide, do not subtract
For 4x = 20, subtracting 4 gives x = 16, which is wrong. The 4 is multiplying, so it must be divided away.
Always check
Put your answer back into the original equation and see whether it balances.
x = 7 in x + 5 = 12 gives 12 = 12. True, so it is right.
Checking costs one line and catches nearly every slip.
Check against the original
Substitute into the equation you were given, not a line you rewrote. A mistake in the rewriting would otherwise go unnoticed.
From words to an equation
Say what the letter stands for, then translate the sentence one piece at a time.
"Seven more than a number is 19" becomes n + 7 = 19.
Watch the order in subtraction: "five less than a number" is n - 5, not 5 - n.
Less than flips the order
"Five less than n" means start at n and go down five: n - 5. Writing 5 - n reverses it, and the two agree only when n is 2.5.
Worked examples
Undoing addition
Solve x + 9 = 23.
- Nine has been added, so subtract nine. The inverse operation.
- Do it to both sides: x + 9 - 9 = 23 - 9. Keeping the balance.
- x = 14. Check: 14 + 9 = 23. True, so the answer is right.
x = 14
Undoing multiplication
Solve 6x = 42.
- Six is multiplying x, so divide by six. Not subtract — the 6 is a multiplier.
- 6x6 = 426. Both sides.
- x = 7. Check: 6 × 7 = 42. Balanced.
x = 7
From words
Four less than a number is 11. Find the number.
- Let n be the number. Say what the letter means first.
- "Four less than n" is n - 4, so n - 4 = 11. Start at n and go down.
- Add four to both sides: n = 15. Check: 15 - 4 = 11. Writing 4 - n = 11 would have given -7.
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Words to know
- Equation
- A statement that two expressions are equal.
- Solution
- The value of the variable that makes the equation true.
- Inverse operation
- The operation that undoes another — subtraction undoes addition.
- Isolate
- To get the variable alone on one side.
- Check
- Substituting your answer back to confirm it works.
Mistakes to watch for
Doing something to only one side. Subtracting five from the left of x + 5 = 12 without doing it to the right leaves x = 12, which does not check.
Subtracting a coefficient instead of dividing. In 4x = 20 the 4 is multiplying, so divide: x = 5. Subtracting gives 16, which fails the check immediately.
Reversing a subtraction in a word problem. "Four less than a number" is n - 4, not 4 - n; the two give answers of 15 and -7.
Skipping the check. Substituting back takes one line and catches almost every arithmetic slip before it matters.
Treating equals as "here comes the answer". It states that two amounts are the same, which is exactly what allows you to work on either side.
The equals sign was invented by Robert Recorde in 1557, who chose two parallel lines because "no two things can be more equal".
The word algebra comes from al-jabr, in the title of a book by al-Khwarizmi around 820 — it means "restoring", as in restoring the balance.
Every spreadsheet, every physics formula and every line of computer arithmetic rests on this one rule about doing the same to both sides.
Problem 1
Solve x + 9 = 23.
Worked answer
14
x = 14
Problem 2
Solve x - 6 = 11.
Worked answer
17
x = 17
Problem 3
Solve 6x = 42.
Worked answer
7
x = 7
Problem 4
Solve x3 = 8.
Worked answer
24
x = 24
Problem 5
Solve x + 12 = 12.
Worked answer
0
x = 0
Problem 6
Solve 9x = 108.
Worked answer
12
x = 12
Problem 7
Solve x - 4.5 = 7.5.
Worked answer
12
x = 12
Problem 8
Solve x5 = 2.4.
Worked answer
12
x = 12
Problem 9
Seven more than a number is 19. Find the number.
Worked answer
12
n = 12
Problem 10
Four less than a number is 11. Find the number.
Worked answer
15
n = 15
Problem 11
Three times a number is 48. Find the number.
Worked answer
16
n = 16
Problem 12
A number divided by 4 is 7.5. Find the number.
Worked answer
30
n = 30
Problem 13
A rectangle has area 60 and width 5. Find its length.
Worked answer
12
L = 12
Problem 14
Is x = 6 a solution of 4x = 20?
- No
- Yes
- Only sometimes
- Cannot tell
Worked answer
No
4 × 6 = 24, not 20
Problem 15
After spending $18, someone has $27 left. How much did they start with?
Worked answer
45
n = 45
Problem 16
Solve 23x = 8.
Worked answer
12
x = 8 × 32 = 12