Solve Two-Step Equations (px + q = r)

Students solve equations of the form px + q = r and p(x + q) = r using properties of equality. They construct simple equations to represent real-world situations and interpret solutions in context.

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Grade 7 · Expressions and Equations · question 1 of 3

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20 rungs. A star at the top.

3(x + 6) = 9, x =

How it is taught

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Undo the add or subtract first.

Then undo the multiply. Both sides.

Read it: Two-step equations and inequalities

4x + 12 = 36 is a tape: 4 equal parts of x and 12 more make 36. Take the 12 away: 4x = 24. Share 24 into 4 equal parts: x = 6.

Undo the last move on x first: − 12 undoes + 12, and ÷ 3 undoes × 3.
Then share what is left into the equal parts: divide both sides.
Check by putting x back in: 4 × 6 + 12 = 36.

Each undo is done to both sides, so the two sides stay equal: 36 − 12 = 24, and 24 ÷ 4 = 6.

An inequality is undone the same way, but dividing both sides by a negative number turns the sign: −2x < 12 becomes x > −6.

x < 6 is every number less than 6: an open circle at 6, because 6 itself is not a solution, and a ray to the left.

Undo in reverse order, on both sides. Dividing an inequality by a negative number turns its sign.

Dividing 4x and 36 by 4 but not the 12: x + 12 = 9, so x = −3.

Take the 12 away first: 4x = 24. Then 24 ÷ 4 = 6, so x = 6.

5x + 6 = 46, x = ___ — 8. Take 6 away from both sides: 5x = 40. Then 40 ÷ 5 = 8, so x = 8.

3x + 5 = 20

Maya buys 3 equal boxes of crayons plus 5 loose crayons. Altogether she has 20 crayons. How many are in each box?

Let x be the crayons per box. We can write this as three x plus five equals twenty.

To get x alone, we undo the operations in reverse order — first the plus five, then the times three.

Subtract five from both sides. Whatever we do to one side, we must do to the other to keep the equation balanced.

On the left, plus five minus five is zero, leaving three x. On the right, twenty minus five is fifteen.

Now undo the times three. Divide both sides by three.

Three x divided by three is just x. Fifteen divided by three is five. So x equals five.

Let's check our answer: three times five is fifteen, plus five is twenty. It works!

Grade 7 · Solve px + q = r by undoing operations

x equals five — there are five crayons in each box. Great work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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Standards

CCSS.7.EE.B.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems

CCSS.7.EE.B.4a

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers

Grade 7

Quoted from the Common Core State Standards.

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A star here is the top of a twenty-rung ladder, in four blocks of equal length — three of facts and one of the skill used in a real question — where a right answer climbs one rung and a wrong one drops back to the start of the block, never below it. The star is kept for good. How we grade

All of Grade 7

This is one rung of 40 in Grade 7 mathematics, in the order they build on each other.

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