One-Step Equations

Study Sheet

One-Step Equations

Undoing what was done to x

Keep the Balance

Concept
The Big Idea

An equation is a balance. Whatever you do to one side you must do to the other, and you undo an operation with its opposite.

An equation is a balance: both sides weigh the same. To find the unknown, undo whatever was done to it, and do the same thing to both sides so the balance stays level. If x+7=12x + 7 = 12, subtract 77 from both sides; if 4x=204x = 20, divide both sides by 44.

Example
Solving 4x=284x = 28
xxxx28
Four equal parts make 28, so one part is 28 ÷ 4 = 7.

xx was multiplied by 44, so divide both sides by 44: x=7x = 7. Check: 4×7=284 \times 7 = 28.

Example
Solving x−6=11x - 6 = 11
116 takenx
Take 6 from x and 11 is left, so x is 11 + 6.

66 was subtracted from xx, so add 66 to both sides: x=11+6=17x = 11 + 6 = 17. Check: 17−6=1117 - 6 = 11.

Side note
Where it goes wrong

Doing the operation to one side only, or undoing with the same operation instead of the inverse (x+7=12x + 7 = 12 needs subtraction, not more addition).

Try it
Try it: two equations

Solve y+14=30y + 14 = 30 and 5y=455y = 45.

Show the answer

Subtract 1414 from both sides: y=30−14=16y = 30 - 14 = \mathbf{16}. Divide both sides by 55: y=45÷5=9y = 45 \div 5 = \mathbf{9}. Checks: 16+14=3016 + 14 = 30 and 5×9=455 \times 9 = 45.

In Plain Terms & More Examples

Concept
In plain terms

Undo whatever was done to the letter using the opposite operation — an inverse operation — doing the same thing to both sides so the balance holds (the properties of equality).

Example
Solving x+9=15x + 9 = 15

Subtract 99 from both sides: x=6x = 6. Check: 6+9=156 + 9 = 15.

Example
Solving x3=4\dfrac{x}{3} = 4
444x
x cut into 3 equal parts of 4: x = 12.

Multiply both sides by 33: x=12x = 12. Check: 12÷3=412 \div 3 = 4.

Side note
Where it goes wrong

Dividing instead of multiplying. In x3=4\dfrac{x}{3} = 4, the xx has been divided by 33, so undo it by multiplying: x=12x = 12. Dividing again gives x=43x = \tfrac43, and then 43÷3\tfrac43 \div 3 is nowhere near 44.

Try it
Try it: undo it

Solve x5=7\dfrac{x}{5} = 7 and 13=x+813 = x + 8.

Show the answer

Multiply both sides by 55: x=35x = \mathbf{35}. Subtract 88 from both sides: x=13−8=5x = 13 - 8 = \mathbf{5}. The unknown can sit on the right; the balance works either way.

Equations with Fractions and Decimals

Concept
Same moves, uglier numbers

To undo multiplication by a fraction, multiply by its reciprocal. To undo a decimal, divide by it — or multiply everything by 1010 first.

Example
Solve 23x=8\tfrac23x = 8
4448
Two thirds of x is 8, so one third is 4 and x is 12.

Multiply both sides by 32\tfrac32: x=8×32=12x = 8\times\tfrac32 = 12. Check: 23×12=8\tfrac23\times12 = 8. ✓

Example
Solving 1.5+x=4.21.5 + x = 4.2

Subtract 1.51.5 from both sides: x=4.2−1.5=2.7x = 4.2 - 1.5 = 2.7. Check: 1.5+2.7=4.21.5 + 2.7 = 4.2.

Side note
Where it goes wrong

Multiplying by the same fraction again. To undo 23x=8\tfrac23x = 8, multiplying by 23\tfrac23 gives 49x=163\tfrac49x = \tfrac{16}{3}, further from xx on its own. Multiply by the reciprocal, 32\tfrac32, because 32×23=1\tfrac32 \times \tfrac23 = 1.

Try it
Try it: solve

Solve x−3.5=7.2x - 3.5 = 7.2 and 0.4x=60.4x = 6.

Show the answer

x=7.2+3.5=10.7x = 7.2 + 3.5 = \mathbf{10.7}. x=6÷0.4=15x = 6 \div 0.4 = \mathbf{15}.

Word Problems

Concept
Four steps
  • Name the unknown with a letter.
  • Write the sentence as an equation.
  • Solve with an inverse operation.
  • Check the answer in the story, not just in the equation.

Checking in the story catches equations that were set up backwards: a number can fit a wrong equation perfectly and still make no sense in the situation.

Example
Setting up

"After spending $18\$18, Ana has $27\$27." Let mm be what she started with: m−18=27m - 18 = 27, so m=45m = 45.

Example
Teams

Some students form 66 teams of 77. How many students are there? Let nn be the number: n6=7\dfrac{n}{6} = 7, so n=7×6=42n = 7 \times 6 = 42.

Side note
Where it goes wrong

Writing the equation backwards. "After spending $18\$18, Ana has $27\$27" is not m+18=27m + 18 = 27: that gives m=9m = 9, and Ana could not spend $18\$18 out of $9\$9. Spending subtracts: m−18=27m - 18 = 27, so m=45m = 45.

Try it
Try it: write and solve

Six identical books weigh 99 kg. Write an equation and find the weight of one book.

Show the answer

6b=96b = 9, so b=1.5b = \mathbf{1.5} kg.

Inequalities

Concept
Many answers at once

An inequality like x>3x > 3 has many solutions: every number greater than 33. Graph it on a number line with an open circle at 33 (not included) and an arrow to the right. For ≥\ge or ≤\le the circle is filled in (included).

An equation pins down a single number; an inequality describes a whole stretch of the line. The circle shows whether the boundary number itself counts, and the arrow shows which way the solutions go.

Example
Graphing x>3x > 3
−1012345678
Open circle: 3 is not included.

Every number to the right of 33 works: 3.13.1, 44, 100100. The number 33 itself does not, because 33 is not greater than 33, so its circle is open.

Example
Graphing x≤−1x \le -1
−6−5−4−3−2−10123
Filled circle: −1 is included.

−1-1 itself works, since −1-1 is equal to −1-1, so its circle is filled. The arrow points left: −1-1, −2-2, −3.5-3.5 and so on.

Example
Solving x+4<10x + 4 < 10

Subtract 44 from both sides, as with an equation: x<6x < 6. Test a number on each side of 66: x=5x = 5 gives 9<109 < 10, true; x=7x = 7 gives 1111, which is not less than 1010.

Side note
Where it goes wrong

Using the wrong circle. "At most 1212 people" allows exactly 1212, so it is n≤12n \le 12 with a filled circle. An open circle would turn away the twelfth person.

Try it
Try it: the ride

A ride needs riders to be at least 4848 inches tall. Write an inequality for a rider's height hh. Can a rider who is 47.547.5 inches tall go on?

Show the answer

"At least 4848" means 4848 or more: h≥48\mathbf{h \ge 48}. The height 47.547.5 is less than 4848, so no.

Why It Works

If two quantities are equal and you change both in the same way, they stay equal — that is what equality means. Subtracting 77 from both sides of x+7=12x + 7 = 12 gives x+7−7=12−7x + 7 - 7 = 12 - 7, and 7−7=07 - 7 = 0 leaves xx alone: x=5x = 5. Checking by substitution (5+7=125 + 7 = 12 ✓) is not optional; it is the proof that the balance was kept at every step.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.