Guide 8A: One-Step Equations
Goal: Get the Variable Alone
Tiny Concept
When we solve an equation, we are trying to answer:
What number makes the equation true?
Example:
x + 5 = 12
We're asking:
What number plus 5 equals 12?
Recipe Box
SADMEP (Undo Operations/Literal Opposite of PEMDAS)
When evaluating expressions:
PEMDAS
When solving equations:
SADMEP
Original Operation | Opposite Operation |
+ | − |
− | + |
× | ÷ |
÷ | × |
One-Step Equation Recipe
Step 1: Find what is attached to the variable.
Step 2: Use the opposite operation.
Step 3: Do it to BOTH sides.
Step 4: Check your answer.
See One:
Solve: x + 5 = 12
Step 1
What is happening to x?
+5
Step 2
Use the opposite.
Opposite of +5 is −5.
Step 3
Subtract 5 from BOTH sides.
x+5−5 = 12−5
x=7
Step 4
Check.
7+5=12 ✓ True
Answer:
x=7
Try One:
Solve: x+9=15
Step 1
What operation is attached to x?
Step 2
What is the opposite operation?
Step 3
Do it to BOTH sides.
x + 9−9 = 15−9
x =______
Check
___ + 9 = 15
x = 6
Do One.
1. x + 4 = 13
2. x+11=20
3. x+17=30
- x = 9
- x = 9
- x = 13
Tiny Reminder
Think of equations like a balance scale.
If you do something to one side:
x+5=12
you MUST do it to the other side to keep the ratio between the two sides the same.
x+5−5 = 12−5
Otherwise the scale tips and you are changing the ratio and integrity of the problem.
Mixed Check
1
What is the value of x?
x+8=19
A) 9
B) 10
C) 11
D) 27
2
What is the value of n?
n+14=84
A) 98
B) 70
C) 68
D) 6
3
Which equation has a solution of 12?
A) x+5=17
B) x+8=24
C) x+10=20
D) x+4=12
4
What number makes the equation true?
m+25=40
A) 15
B) 25
C) 65
D) 10
5
A number increased by 7 equals 18.
Which equation matches the situation?
A) x−7=18
B) x+7=18
C) 7−x=18
D) 18+7=x
Answer Key
Do One
- 9
- 9
- 13
Mixed Check
- C
- B
- A
- A
- B
Guide 8B: Two-Step Equations
Goal: Undo TWO Things To Get The Variable Alone
Tiny Concept
In a one-step equation, only ONE thing is attached to the variable.
Example: x+5=12
In a two-step equation, TWO things are attached to the variable.
Example: 3x+5=17
The variable is:
- multiplied by 3
- then increased by 5
To solve:
Undo things in reverse order.
Recipe Box
SADMEP
When solving equations:
Attached to Variable | Undo With |
+ | − |
− | + |
× | ÷ |
÷ | × |
Two-Step Recipe
Step 1: Find the variable.
Step 2: Undo addition or subtraction first.
Step 3: Undo multiplication or division second.
Step 4: Check your answer.
Important Idea
Think: How was the variable built?
3x+5
- Started as x → x
- Multiply by 3 → 3x
- Add 5 → 3x+5
When solving: Go backwards.
- Subtract 5.
- Then divide by 3.
See One
Solve: 3x+5=17
Step 1
Undo +5.
Subtract 5 from both sides.
3x+5−5 = 17−5
→ 3x=12
Step 2
Undo ×3.
Divide both sides by 3.
33x=312
→ x=4
Step 3
Check.
3(4)+5
→ 12+5=17 ✓ True
Answer:
x=4
Try One
Solve: 2x+7=19
Step 1
Undo ______
Subtract ____ from both sides.
should get → 2x=12
Step 2
Undo ______
Divide both sides by ____.
x=_____
Check
2(___)+7=19
x=6
Do One
1. 4x+3=19
2. 5x+8=33
3. 6x+4=40
Mixed Check
1
What is the value of x?
2x+3=15
A) 5
B) 6
C) 9
D) 12
2
What is the value of n?
4n+6=26
A) 5
B) 6
C) 8
D) 20
3
What is the value of y?
3y+9=24
A) 3
B) 5
C) 6
D) 8
4
A number multiplied by 7 and then increased by 4 equals 32.
Which equation matches?
A) 7x - 4=32
B) 7+x+4=32
C) 4x+7=32
D) 7x + 4=32
5
Which step should happen first when solving
5x+8=23
A) Divide by 5
B) Multiply by 5
C) Subtract 8
D) Add 8
Common Mistakes
Mistake #1
3x+5=17
Student divides by 3 first.
❌ Wrong
The +5 is still attached.
Always undo addition/subtraction first.
Mistake #2
Forgetting BOTH sides.
3x+5=17
Subtract 5 from the left.
Forget the right.
❌ Wrong
Must stay balanced.
Mistake #3
Not checking.
Always plug the answer back in. (even mentally if going for speed)
Tiny Reminder
The variable is trapped.
Look at what is trapping it.
Remove the traps in reverse order.
3x+5
Last thing added:
+5
Remove it first.
Then remove:
×3
Guide 8C: Variables on Both Sides
Goal: Get All the Variables Together and then isolate the variable.
Tiny Concept
Sometimes the variable appears on BOTH sides of the equation.
Example: 5x+2=2x+14
You cannot solve until all the x's are together.
Think:
"Let's get all the x's onto one team."
Then:
"Let's get all the regular numbers onto the other team."
:. in other words combine your like terms (variables, constants)
Recipe Box
Variables on Both Sides Recipe
Step 1: Move variables together/combine like terms even across the equal sign.
Step 2: Move constants together/combine like terms even across the equal sign.
Step 3: Solve.
Step 4: Check.
Important Idea: Combining Like Terms
Variables go together.
Numbers go together.
Example:
5x+2=2x+14
5x and 2x (like terms/variables)
2 and 14 (like terms/constants)
See One
Step 1:
Move variables together.
Move the smaller variable term. (2x < 5x)
Subtract 2x from both sides.
5x+2 - 2x = 2x + 14 - 2x
since 5x - 2x = 3x
→ 3x + 2 = 14
Step 2:
Move constants together.
Subtract 2 from both sides.
3x + 2 - 2 = 14 - 2
since 14 - 2 = 12
→ 3x = 12
Step 3:
Divide by 3.
33x = 312
since 12÷3 = 4
→ x=4
Step 4:
Check.
5(4)+2=2(4)+14
→ 20+2=8+14
→ 22=22 ✓
x=4
Try One
Solve: 4x+3=x+12
Step 1
Move variables together.
Subtract ___ from both sides.
4x + 3 - ____= x + 12 - _____
→ ___x + 3 = 12
Step 2
Move constants together.
Subtract 3 from both sides.
3x + 3 - 3 = 12 - 3
→ _____ = 9
Step 3
Divide ___ from both sides.
33x = 39
→ x=____
Check
4(___) + 3 = ____+12
_____ = 15 ✓
x=3
Do One
1. 6x+5=3x+20
2. 8x+4=5x+19
3. 7x+2=4x+23
Mixed Check
1
What is the value of x?
3x+4=x+14
A) 3
B) 4
C) 5
D) 7
2
What is the value of n?
5n+7=2n+19
A) 3
B) 4
C) 5
D) 6
3
What is the value of y?
6y+2=4y+12
A) 3
B) 4
C) 5
D) 6
4
A student starts with:
7x+1=4x+16
What should they do FIRST?
A) Add 16
B) Divide by 7
C) Subtract 4x
D) Subtract 1
5
Which equation has a solution of x=5?
A) 4x+2=x+17
B) 4x+2=x+20
C) 4x+2=x+12
D) 4x+2=x+8
Common Mistakes
Mistake #1
Moving numbers before variables.
5x+2=2x+14
You CAN do it, it’s just not encouraged.
But it usually makes the problem harder.
Get the variables together first.
Mistake #2
Subtracting the larger variable term.
5x+2=2x+14
If you subtract 5x:
2=−3x+14
This works, but introduces negatives.
Usually move the smaller variable term.
Mistake #3
Only moving one side.
Remember:
Whatever happens to one side must happen to the other side.
Tiny Reminder
Variables Together
5x+2=2x+14
↓
3x+2=14
Numbers Together
3x=12
Solve
x=4
Variables together → Numbers together → Solve
Beyond This Unit: SAT Applications
Why This Matters for the SAT
The SAT almost never asks:
3x+5=17 by itself.
Instead, it hides equation solving inside:
- word problems
- geometry
- systems
- quadratics
- functions
- data analysis
If you cannot solve equations quickly, harder SAT questions become impossible.
SAT Skill #1
Direct Equation Solving (Easy)
What value of x satisfies
4x+7=31
This is essentially a two-step equation.
Difficulty:
⭐ Easy
SAT Skill #2
Equation Hidden in a Word Problem (Medium)
A gym charges a one-time registration fee of $25 and $15 per month.
If a member paid $130 total, how many months was the membership active?
Translate:
15m+25=130
Now solve.
Difficulty:
⭐⭐ Medium
SAT Skill #3
Variables on Both Sides + Layered Solving (Medium)
The equation
5k+12=2k+30 has solution k.
What is the value of k - 7?
They want you to solve for k using what you know about variables on both sides AND then ask you to do something with that solution.
Difficulty:
⭐⭐ Medium
SAT Skill #4
Equation Embedded in Geometry (Hard)
The perimeter of a rectangle is 42.
One side is 2x+1 and another side is x+5
Find x.
Now students must:
- Build the perimeter equation:
- length + length + width + width = perimeter
- 2L + 2W = P
- Understand that:
- length = one side = 2x +1
- width = one side = x + 5
- SO…. (2)(2x + 1) + (2)(x+5) = 42
- Solve a multi-step equation
OR
Difficulty:
⭐⭐⭐ Hard
SAT Skill #5
Equation Hidden Inside a Function (Hard)
If f(x) = 3x+8 and f(x)=29
what is x?
Student must recognize they equal the same thing and thus equal each other:
3x+8=29
Difficulty:
⭐⭐⭐ Hard
SAT Wording Watch
The SAT rarely says:
Solve for x.
Instead it says:
- What is the value of x?
- Which value satisfies the equation?
- What is the value of the constant?
- What is the number of months?
- What is the side length?
- What is the solution?
- For what value of x?
- If f(x)=___ and f(x)=___, what is x?
SAT Trap Watch
Trap #1
Students know how to solve.
But don't recognize an equation is hiding.
Example:
A taxi charges $4 plus $3 per mile. A ride cost $40.
Many students don’t know to build:
3m+4=40
Trap #2
Variables on both sides.
Many students immediately panic:
5x+12=2x+30
even though it is the exact same recipe as 5x + 12 = 30 or 5 - x = 10; there are just more simple operational steps.
Trap #3
Fractions.
SAT loves:
3x+5 = 17
Same skills.
Different appearance.