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Expressions and Equations 6-8 / Week 06 / Monday
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Week 06 · Two-Step Equations

Monday

Two envelopes and some beans
// Undo in reverse order, rewrite to see the puzzle, estimate to check
⏱ about 20 min

Monday: Two Envelopes and Some Beans

Rocket sets two sealed envelopes in the left cup and adds 3 loose beans. "Both envelopes hide the same number."

Raven spoons beans into the right cup, one at a time, until the hanger settles. "Eleven," she counts.

"So 2x + 3 = 11," Rocket says. "Two things happened to x. It got doubled, then 3 got added."

"What do you notice about the order?" Raven asks. "Which one should we undo first?"

Rocket reaches for the envelopes, then stops. "The 3 was added last. So... take the 3 off first?"

Nova hovers above the hook, her light steady. "Reverse order," she says. "Last on, first off."

Raven lifts 3 beans from each cup. Two envelopes face 8 beans. Rocket grins. "Now share." Nova hums. "Then check."

Rocket adds envelopes to one cup of the coat-hanger balance while Raven spoons beans into the other, Nova above.

Two steps were done, so two steps undo

In 2x + 3 = 11, two things were done to x. First it was multiplied by 2. Then 3 was added.

To get x alone, undo them in reverse order. The 3 was added last, so subtract 3 first. Then divide by 2.

Do each undo to both sides. The balance stays level the whole way.

A level balance: two envelopes and 3 beans on the left, 11 beans on the right.
  1. Write the equation: 2x + 3 = 11.
  2. Subtract 3 from both sides: 2x = 8.
  3. Divide both sides by 2: x = 4.
  4. Check: 2 × 4 + 3 = 11. True.
A level balance after taking 3 beans from each side: two envelopes on the left, 8 beans on the right.
A level balance after sharing: one envelope on the left, 4 beans on the right.

Undoing in the wrong order makes a mess. Dividing 2x + 3 = 11 by 2 first gives x + 1.5 = 5.5, which still works but is harder.

The numbers can be decimals or fractions. 0.5x + 3 = 7 undoes the same way: subtract 3, then divide by 0.5.

TEST BY SUBSTITUTION
  • Read the question.
  • Tap your answer.
Which value of x makes 2x + 3 = 11 true: 3, 4, 7?
Which value of x makes 3x - 1 = 14 true: 4, 5, 13?
Which value of x makes 0.5x + 3 = 7 true: 4, 8, 20?
UNDO IN REVERSE ORDER
  • Read the question.
  • Tap your answer.
What is x if 2x + 3 = 11?
What is x if 3x - 1 = 14?
What is x if 0.5x + 3 = 7?
What is x if 4x + 6 = 9?
StatementTrue or false?
x = 4 is a solution of 2x + 3 = 11.?
x = 7 is a solution of 2x + 3 = 11.?
In 2x + 3 = 11, the 3 was added last, so it is undone first.?
x = 5 is a solution of 3x - 1 = 14.?
WHY THIS EXERCISEx = 4 makes 2 × 4 + 3 = 11 true. Dividing before subtracting still works, but subtracting first keeps the numbers whole.
How many beans are in each envelope if 2x + 3 = 11? Type the number.
WHY THIS EXERCISE2 × 4 + 3 = 11 is true, so each envelope holds 4 beans.
Try it
On paper, draw a balance with three envelopes and 2 beans on the left and 14 beans on the right.
Write the equation. Undo the 2 first, then the 3. Check by substitution.
Draw 2x + 3 = 11 as three balance pictures: the start, after taking 3 from each side, and after sharing.

Two steps, undone in order. Tomorrow a group of beans and envelopes is multiplied all at once: p(x + q) = r.