Two pages of visual notes for students to keep: the key words with pictures, the steps for solving any equation, four solved problems in the style of the Tip the Scales game, and where equations show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Solving Equations
Grades 6–8 · CCSS 6.EE.B.7, 7.EE.B.4a, 8.EE.C.7b · Pairs with the Tip the Scales game
Name
The big idea: an equation is a balance. Both sides weigh the same. To find x, take things off or split things up, but whatever you do to one side, do to the other, so it stays balanced. When x is alone, you've solved it.
Key vocabulary (in 2x + 3 = 11)
Equation
2x + 3 = 11
Two expressions that are equal; it has an = sign.
Variable
2x + 3 = 11
A letter that stands for an unknown number.
Coefficient
2x + 3 = 11
The number multiplied by the variable.
Constant
2x + 3 = 11
A number on its own, with no variable.
Solution
x = 4
The value that makes it true: 2(4) + 3 = 11 ✓
Inverse operations
+ ↔ − × ↔ ÷
Operations that undo each other.
Like terms
3x + 5x = 8x
Terms with the same variable part; combine them.
Distributive property
2(x + 4) = 2x + 8: multiply each term inside.
The steps, in order (skip any step you don't need)
1. DistributeClear parentheses: 3(x + 2) → 3x + 6
2. CombineLike terms on each side: 2x + 5x → 7x
3. Gather xMove x-terms to one side: subtract the smaller one
4. Undo + or −Add or subtract the constant from both sides
5. Undo × or ÷Divide (or multiply) both sides. Then check!
Why undo in that order? It's the order of operations backwards. In 2x + 3, x is multiplied first and 3 is added last, so to free x, take off the 3 first, then divide by 2. Like getting dressed and undressed: shoes come off before socks.
Gather x: subtract 2x from both sides: 4x + 1 = 13
Undo + 1: 4x = 12
Undo × 4:x = 3
✓ Check: 6(3) + 1 = 19 and 2(3) + 13 = 19
3. The distributive property
3(x + 4) = 27
Distribute: 3x + 12 = 27
Undo + 12: 3x = 15
Undo × 3:x = 5
✓ Or divide first: x + 4 = 9, so x = 5
4. Real life: at the movies
A ticket is $12 and each snack is $4. You spent $24. How many snacks?
Write it: 12 + 4x = 24 (x = number of snacks)
Undo + 12: 4x = 12
Undo × 4:x = 3 snacks
✓ Check: $12 + 3 × $4 = $24
Watch out
Do the same thing to both sides. Changing only one side tips the scale.
Distribute to every term: 3(x + 4) is 3x + 12, not 3x + 4.
Watch the signs: subtracting a negative is adding. Then always check your answer.
Where you see it in real life
Scales: balanced means both sides are equal.Shopping: work back from a total to find a price.Taxis: $3 + $2 per mile = $15. How many miles?Prices: 3x = 12, so each costs $4.