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Expressions and Equations 6-8 / Week 04 / Tuesday
2/6
Week 04 · The Balance: One-Step Equations

Tuesday

Do the same to both sides
// An equation is a level balance, and a solution keeps it level
⏱ about 20 min

Tuesday: Do the Same to Both Sides

Raven lifts 3 beans out of the left cup. The balance tips toward the right. "Now it is not an equation."

"Take 3 from the other side too," Rocket says, scooping them out. The hanger swings, then settles level again.

"What do you notice?" Raven asks. "The envelope is alone now, and 5 beans sit across from it."

"So the envelope holds 5," Rocket says. "We did not guess. We undid the plus 3."

Nova drifts lower, her glow bright. "Whatever you do to one side, do to the other," she says. "That keeps the balance."

Rocket sets three envelopes in the left cup and 12 beans in the right. "Three equal envelopes. What undoes that?"

Nova hums. "Think about sharing the beans fairly."

Undo with the inverse operation

Every operation has an inverse that undoes it. Subtracting 3 undoes adding 3. Dividing by 3 undoes multiplying by 3.

To get the letter alone, undo what was done to it. Do the same thing to both sides, and the equation stays true.

EquationWhat was done to xUndo it on both sidesSolution
x + 3 = 8add 3subtract 3x = 5
x - 4 = 9subtract 4add 4x = 13
3x = 12multiply by 3divide by 3x = 4
x ÷ 2 = 7divide by 2multiply by 2x = 14

x + p = q: take away the beans you can see

A level balance: an envelope and 3 beans on the left, 8 beans on the right.
A level balance after taking 3 beans from each side: the envelope alone on the left, 5 beans on the right.
  1. Write the equation: x + 3 = 8.
  2. Subtract 3 from both sides: x + 3 - 3 = 8 - 3.
  3. Simplify: x = 5.
  4. Check by substitution: 5 + 3 = 8. True.

px = q: share the beans among the envelopes

A level balance: 3 envelopes on the left, 12 beans on the right.
A level balance after sharing: one envelope on the left, 4 beans on the right.

3x = 12 means three equal envelopes balance 12 beans. Divide both sides by 3 to find one envelope.

3x ÷ 3 = 12 ÷ 3, so x = 4. Check: 3 × 4 = 12. True.

The numbers can be fractions or decimals too. If x + 1.5 = 4, subtract 1.5 from both sides: x = 2.5.

TAKE AWAY ON BOTH SIDES
  • Read the question.
  • Tap your answer.
What is x if x + 3 = 8?
What is x if x + 9 = 21?
What is x if x + 1.5 = 4?
What is x if x + 2/3 = 2?
SHARE ON BOTH SIDES
  • Read the question.
  • Tap your answer.
What is x if 3x = 12?
What is x if 5x = 45?
What is x if 4x = 10?
What is x if 0.5x = 7?
SOLVE 4X = 28, STEP BY STEP
  • ?Ask what was done to x: it was multiplied by 4.
  • ?Divide both sides by 4.
  • ?Write the equation: 4x = 28.
  • ?Simplify: x = 7.
  • ?Check: 4 × 7 = 28.
WHY THIS EXERCISEUndoing the operation on both sides finds the solution, and the check proves it is right.
Solve x - 4 = 9. What is x?
Solve x ÷ 2 = 7. What is x?
Try it
Write x + 6 = 15 on paper. Under it, write the undo step and the solution.
Then write 6x = 15 and solve it the same way. Say which answer is a whole number and which is not.
Draw 3x = 12 as two balance pictures: three envelopes against 12 beans, then one envelope against its share.

Two undo rules, both checked. Tomorrow is Puzzle Lab: you build the balance and hide your own numbers.

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