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

Friday

Equations all around
// Envelopes on both sides
⏱ about 20 min

Friday: Equations All Around

Rocket paces the garage. "I read 20 pages a day. Raven started 30 pages ahead and reads 15 a day."

"When do we land on the same page?" He grabs a card. "20d = 15d + 30."

"What do you notice about the d terms?" Raven asks.

"Take 15d off both sides. 5d = 30. So d is 6. Day six, same page."

Nova hovers over the card, her light tapping it twice. "And if you both read 20 pages a day?"

"Then 20d = 20d + 30," Rocket says. "0 = 30. Never. She stays 30 pages ahead forever."

"That is the no-solution card in real life," Raven says. "Equal rates never catch up."

"Puzzle Night is in two weeks," Rocket says. "I am saving that one for the family."

Where both-sides equations live

Catching up: 20d = 15d + 30 asks when two readers reach the same page. Solving gives d = 6.

Equal rates: 20d = 20d + 30 simplifies to 0 = 30. The readers never meet. No solution.

Sharing beans: 3 cups with x beans each plus 4 equals 2 cups plus 10. Then 3x + 4 = 2x + 10, so x = 6.

Water levels: one jug starts at 2 cm and rises 3 cm a minute. Another starts at 8 cm and rises 1 cm a minute. They are level when 3m + 2 = m + 8, so m = 3 minutes.

The week in five lines

  • A linear equation can have x on both sides. Take the same x term off both sides to collect it.
  • The five moves: distribute, collect like terms, letters to one side, numbers to the other, divide.
  • Fraction and decimal coefficients follow the same moves. Multiplying through can clear a fraction first.
  • Three endings: x = a (one solution), a = a (infinitely many), a = b (no solution).
  • Always check by putting the answer back into the original equation.
EVERYDAY EQUATIONS
  • Read the question.
  • Tap your answer.
Rocket reads 20 pages a day. Raven is 30 pages ahead and reads 15 a day. After how many days are they on the same page?
One jug is at 2 cm and rises 3 cm a minute. Another is at 8 cm and rises 1 cm a minute. After how many minutes are they level?
Rocket has 4 cups of x beans plus 6 loose beans. Raven has 6 cups of x beans. The piles match. What is x?
Both readers read 20 pages a day, and Raven is 30 pages ahead. How many solutions does 20d = 20d + 30 have?
MIXED REVIEW
  • Read the question.
  • Tap your answer.
What is x if 2(x + 5) = 3x + 4?
What is x if 0.4x + 2 = 0.2x + 5?
How many solutions does 2(3x + 1) = 6x + 2 have?
Check: if x = 6, what is 3x + 4?
StatementTrue or false?
To collect x terms on one side, subtract the same x term from both sides.?
2(x + 3) is equivalent to 2x + 6.?
3(x - 1) is equivalent to 3x - 1.?
20d = 20d + 30 has no solution.?
6 is a solution of 2(x + 5) = 3x + 4.?
WHY THIS EXERCISEThese five lines are the whole week. Next week two equations run at the same time.
Try it
Pick two things at home that grow at different rates, like two drips or two counters.
Give one a head start. Write the catch-up equation and solve it. Then watch to check.
Keep water over a tray or the sink, and wipe spills right away.
Draw the two readers as two number lines, pages marked. Show where day 6 lands on each.

Terrific week. You can solve any linear equation and name its ending. Tomorrow is Puzzle Day.

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