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

Monday

A level balance
// An equation is a level balance, and a solution keeps it level
⏱ about 20 min

Monday: A Level Balance

Rocket hooks a wire coat hanger on the garage shelf and ties a paper cup to each end. "The Puzzle Post balance."

He drops an envelope and 3 beans into the left cup, then counts 8 beans into the right. The hanger settles level.

"The envelope hides some beans," Rocket says. "I am not telling how many."

Raven studies the two cups. "What do you notice? Both sides weigh the same, so both sides count the same."

Nova hovers above the hook, her light steady. "Then write what you see," she says. "An envelope, plus 3, equals 8."

"x + 3 = 8," Raven writes. "An equation."

Rocket grins. "So x is... no, wait. Let me test it before I shout."

Nova hums. "Test a number. The balance will tell you."

Rocket and Raven at the workbench with a coat-hanger balance, an envelope in one cup, beans in both, Nova hovering.

An equation is a level balance

An equation is a number sentence with an equals sign. It says the two sides name the same number.

x + 3 = 8 is an equation. The left side is the cup with the envelope and 3 beans. The right side is the cup with 8 beans.

The hanger is level, so the two cups hold the same number of beans. The envelope must hide exactly enough to make that true.

A level balance: an envelope and 3 beans on the left, 8 beans on the right.

A solution makes it true

A solution of an equation is a value of the letter that makes the equation true. You find out by substitution: put the number in for x and check.

Try x = 4 in x + 3 = 8. The left side becomes 4 + 3 = 7. That is not 8, so 4 is not a solution.

Try x = 5. The left side becomes 5 + 3 = 8. Both sides are 8. The balance stays level, so x = 5 is the solution.

A one-letter equation like this has exactly one solution. Every other value tips the balance.

A tipped balance: the envelope holds the wrong number of beans, so the cups do not match.
TEST BY SUBSTITUTION
  • Read the question.
  • Tap your answer.
Which value of x makes x + 3 = 8 true: 3, 4, 5?
Which value of x makes x + 6 = 10 true: 2, 4, 6?
Which value of x makes 2x = 14 true: 5, 7, 12?
Which value of x makes x - 4 = 9 true: 5, 13, 14?
PUT THE NUMBER IN
  • Read the question.
  • Tap your answer.
If the envelope holds 7 beans, how many beans does x + 3 count?
Four envelopes each hold 6 beans. How many beans is 4x?
If x = 1.5, what is x + 2.5?
StatementTrue or false?
x = 5 is a solution of x + 3 = 8.?
x = 11 is a solution of x + 3 = 8.?
x = 4 is a solution of 3x = 12.?
x = 4 is a solution of x - 2 = 6.?
WHY THIS EXERCISESubstitution is the test. A solution makes both sides the same number, so the balance stays level.
What number is in the envelope if x + 3 = 8? Type the number.
WHY THIS EXERCISE5 + 3 = 8 is true, and no other number works, so x = 5 is the solution.
Try it
On paper, draw a balance with an envelope and 4 beans on the left and 10 beans on the right.
Write the equation under it. Test x = 5, then x = 6. Circle the one that keeps it level.
Draw the balance level with x + 3 on the left and 8 on the right. Then draw it tipped with x = 4.

A level start. Tomorrow you learn to find the hidden number without guessing: do the same to both sides.