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Expressions and Equations 6-8 / Week 10 / Monday
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Week 10 · Solving Linear Equations

Monday

Envelopes on both sides
// Envelopes on both sides
⏱ about 20 min

Monday: Envelopes on Both Sides

Rocket hangs the coat hanger on the garage hook and loads both cups. "New puzzle. Envelopes on both sides."

The left cup holds two envelopes and three beans. The right cup holds one envelope and eight beans. The hanger is level.

"Every envelope hides the same number of beans," Rocket says. "Guess it."

Raven studies both cups. "What do you notice that is the same on both sides?"

"One envelope on each side," Rocket says. "If I lift one envelope off each cup, it stays level."

Nova hovers over the left cup, her light on the beans. "Now what is left?" she asks.

"One envelope and three beans against eight beans," Raven says. "So the envelope hides five."

Rocket opens an envelope. Five beans. "The balance did the thinking."

Rocket, Raven and Nova at the Puzzle Post with the coat-hanger balance loaded with envelopes and beans on both sides.

The letter on both sides

The balance shows an equation. Left cup: 2x + 3. Right cup: x + 8. Level means 2x + 3 = x + 8.

Each envelope is an x, the same unknown number of beans. This time x sits on both sides.

The trick is the same as before. Whatever you take off one cup, take off the other too.

Take one envelope off each side: x + 3 = 8. Take three beans off each side: x = 5.

A level coat-hanger balance: two envelopes and three beans on the left, one envelope and eight beans on the right.

Check by substitution

Put 5 in for every x. Left: 2 × 5 + 3 = 13. Right: 5 + 8 = 13. Both sides agree.

A value that makes both sides equal is a solution. Checking is part of solving, not an extra.

Terms with the same letter part are like terms. 2x and x are like terms, so you can collect them.

  1. Write the equation the balance shows, one side for each cup.
  2. Take the smaller number of envelopes off both sides.
  3. Take the smaller number of beans off both sides.
  4. If envelopes remain, share the beans equally among them.
  5. Check by putting your answer back into the original equation.
READ THE BALANCE
  • Read the question.
  • Tap your answer.
A coat-hanger balance, level: the left cup holds 2 envelopes marked x and 3 beans, the right cup holds 1 envelope marked x and 8 beansThe balance shows 2x + 3 = x + 8. How many beans hide in each envelope?
A coat-hanger balance, level: the left cup holds 3 envelopes marked x and 2 beans, the right cup holds 1 envelope marked x and 10 beansThe balance shows 3x + 2 = x + 10. How many beans hide in each envelope?
A coat-hanger balance, level: the left cup holds 2 envelopes marked x and 6 beans, the right cup holds 4 envelopes marked xThe balance shows 2x + 6 = 4x. How many beans hide in each envelope?
WHICH VALUE FITS?
  • Read the question.
  • Tap your answer.
Which value of x makes 3x + 1 = 2x + 7 true: 4, 6, 8?
Which value of x makes 5x + 2 = 3x + 10 true: 3, 4, 5?
Which value of x makes 2x + 9 = 4x + 1 true: 2, 4, 6?
StatementTrue or false?
Taking one envelope off both cups keeps the balance level.?
3x and 5 are like terms.?
5 is a solution of 2x + 3 = x + 8.?
6 is a solution of 3x + 2 = x + 10.?
2 × 5 + 3 = 5 + 8?
WHY THIS EXERCISEA solution is checked, not guessed. The balance only tells the truth when both sides agree.
Try it
Hang your balance and load two envelopes and three beans against one envelope and eight beans.
Lift one envelope off each cup. Then lift three beans off each cup. What is left tells you x.
Hang the hanger at chest height on a strong hook a grown-up has checked.
Draw a balance for 3x + 2 = x + 10 with envelopes for x. Cross off one envelope on each side, then two beans on each side.

Strong start. Tomorrow the equations get parentheses, fractions and decimals, and you learn the full method.