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Unit 6 · Standard 6.AT.7

Identify Equivalent Algebraic Expressions

Lesson 6-6 · Learn It — how the math works

Why we're learning this

At the music studio, two producers wrote formulas for the cost of renting equipment. Producer A wrote 3x + 2x + 10, and Producer B wrote 5x + 10. They look different, but are they really? Let's check: if x = 4, Producer A gets 3(4) + 2(4) + 10 = 30, and Producer B gets 5(4) + 10 = 30. Same result — they're equivalent expressions!

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When are two expressions equivalent?

Two expressions are equivalent when they always give the same value, no matter what number you put in for the variable. They can look different but mean the same thing.

Picture itSee what the math looks like before you read about it

The same square is shown two ways — broken into green strips and yellow unit squares on the left, and as one whole red square on the right.
Where you see this in real lifeThe same square is shown two ways — broken into green strips and yellow unit squares on the left, and as one whole red square on the right.

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram at x = 4: Producer A's 3x + 2x + 10 is 12 + 8 + 10 = 30, and Producer B's 5x + 10 is 20 + 10 = 30.30A123x82x101030B205x1010
  1. Producer A wrote 3x + 2x + 10 and Producer B wrote 5x + 10. Are they ?
  2. I test x = 4. For A: 3(4) + 2(4) + 10 = 12 + 8 + 10 = 30.
  3. For B: 5(4) + 10 = 20 + 10 = 30.
  4. Both give 30, and 3x + 2x is just 5x, so they are .

Watch out — mistakes students make here

  • 9n is right, but the constants are 4 − 1 = 3, not 5. The −1 was added instead of subtracted.
  • 18n multiplies the coefficients (6 × 3) instead of adding them, and the −1 was left out. Add like terms: 6n + 3n = 9n, and 4 − 1 = 3.
  • 63n writes the 6 and the 3 side by side and drops the constants. Add the coefficients, 6n + 3n = 9n, then combine 4 − 1 = 3.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's check 2x + 3x and 5x. Try x = 2.
  2. 2(2) + 3(2) = 4 + 6 = 10, and 5(2) = 10. Same answer.
  3. So 2x + 3x and 5x are .

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Sort each expression into the group with its match.
  • or test a value to check.

Solve: Which expression is to 4x + 3x?

My answer

Stuck? Open one hint at a time.

Hint 1

4x and 3x share the exact same variable, x — that makes them like terms.

Hint 2

Combine like terms by adding the coefficients (the number parts). The variable tags along unchanged.

Hint 3

Add the coefficients: 4 + 3. Keep the x exactly as it is — no new exponent appears.

Check my answer
✅ Answer: 7x
4x + 3x = 7x. Combine the coefficients: 4 + 3 = 7.

Turn & TalkSay it out loud with a partner

Producer A wrote 3x + 2x + 10 and Producer B wrote 5x + 10. They look different but both give 30 when x = 4. Why are they ?

Try starting with:

  • These expressions are equivalent because 3x + 2x ___.Estas expresiones son equivalentes porque 3x + 2x ___.
  • I checked by substituting x = ___ and getting ___ both ways.Lo comprobé sustituyendo x = ___ y obteniendo ___ en ambas.
Use these wordsEquivalentSimplifyLike TermsCombine