Isolating quantities

Kevin Lee
6 min read
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Study Guide Overview
This study guide covers variable isolation for SAT Math. It explains using inverse operations (SADMEP), combining like terms, and the distributive property to solve equations. It also includes strategies for solving equations with variables on both sides and provides practice questions.
AP SAT Math: Mastering Variable Isolation ๐
Hey there! Ready to ace the SAT Math section? Let's dive into isolating variables โ a skill that's absolutely crucial for solving equations quickly and accurately. Think of it as detective work: we're uncovering the value of a variable by strategically using math operations. This guide will be your go-to resource the night before the exam, making sure you're confident and ready to tackle any equation!
Isolating Variables with Inverse Operations
Understanding Inverse Operations
Isolating a variable means getting it all alone on one side of the equation. We do this by using inverse operations โ the opposite of whatever's happening to the variable.
- Addition and subtraction are inverse operations.
- To undo addition, subtract from both sides.
- To undo subtraction, add to both sides.
- Multiplication and division are inverse operations.
- To undo multiplication, divide both sides.
- To undo division, multiply both sides.
- Remember to always perform the same operation on both sides of the equation to maintain balance.
- Think of it like a seesaw โ if you add weight to one side, you have to add the same weight to the other side to keep it balanced.
- Use the reverse order of operations (SADMEP - Subtraction, Addition, Division, Multiplication, Exponents, Parentheses) when isolating a variable.
Examples of Isolating Variables
- Example 1: Isolating in
- Subtract 5 from both sides:
- Simplify:
- Example 2: Isolating in
3y = 21
- Divide both sides by 3:
3y รท 3 = 21 รท 3
- Simplify:
- Divide both sides by 3:
Combining Like Terms for Isolation
Identifying and Combining Like Terms
- Like terms have the same variable raised to the same power (e.g.,
2x
and3x
, but not2x
and3x^2
). - Combine like terms by adding or subtracting their coefficients (the numbers in front of the variables).
- Combining like terms simplifies the equation before you isolate the variable.
Examples of Combining Like Terms
- Example 1: Simplifying
2x + 3x + 5 = 20
- Combine like terms:
5x + 5 = 20
- Subtract 5 from both sides:
5x = 15
- Divide both sides by 5:
- Combine like terms:
- Example 2: Simplifying
4y - 2y + 7 = 13
- Combine like terms:
2y + 7 = 13
- Subtract 7 from both sides:
2y = 6
- Divide both sides by 2:
- Combine like terms:
Isolating Variables with the Distributive Property
Applying the Distributive Property
- The distributive property states that . Basically, you multiply the term outside the parentheses by each term inside.
- Use this property to eliminate parentheses and then combine like terms.
Examples of Using the Distributive Property
- Example 1: Solving
3(x + 2) = 15
- Distribute 3:
3x + 6 = 15
- Subtract 6 from both sides:
3x = 9
- Divide both sides by 3:
- Distribute 3:
- Example 2: Solving
2(y - 4) + 5 = 17
- Distribute 2:
2y - 8 + 5 = 17
- Combine like terms:
2y - 3 = 17
- Add 3 to both sides:
2y = 20
- Divide both sides by 2:
- Distribute 2:
Solving Equations with Variables on Both Sides
Strategies for Variables on Both Sides
- When variables appear on both sides of the equation, the goal is to gather all variable terms to one side and constant terms to the other.
- Use inverse operations to move variable terms from one side to the other, and then combine like terms.
Examples of Solving with Variables on Both Sides
- Example 1: Solving
3x + 5 = x + 15
- Subtract from both sides:
2x + 5 = 15
- Subtract 5 from both sides:
2x = 10
- Divide both sides by 2:
- Subtract from both sides:
- Example 2: Solving
4y - 3 = 2y + 7
- Subtract
2y
from both sides:2y - 3 = 7
- Add 3 to both sides:
2y = 10
- Divide both sides by 2:
- Subtract
Remember SADMEP (Subtraction, Addition, Division, Multiplication, Exponents, Parentheses) โ it's PEMDAS in reverse! Use this order to undo operations and isolate variables.
Final Exam Focus ๐ฏ
Key Areas:
- Isolating variables using inverse operations.
- Combining like terms to simplify equations.
- Applying the distributive property effectively.
- Solving equations with variables on both sides.
Time Management:
- Don't spend too long on any one question. If you're stuck, move on and come back later.
- Practice solving equations quickly to build your speed and confidence.
Common Pitfalls:
- Forgetting to perform operations on both sides of the equation.
- Incorrectly applying the distributive property.
- Making errors when combining like terms.
- Not using SADMEP (reverse order of operations) when isolating variables.
Practice Questions
Practice Question
Multiple Choice Questions
-
Solve for :
5x - 7 = 23
(A) 3 (B) 4 (C) 6 (D) 8 -
Solve for :
2(y + 3) = 16
(A) 2 (B) 5 (C) 8 (D) 10 -
Solve for :
4z + 9 = z + 18
(A) 2 (B) 3 (C) 6 (D) 9
Free Response Question
Solve the following equation for : 3(2x - 1) + 5 = 2x + 10
Scoring Breakdown:
Step | Points |
---|---|
Apply the distributive property: 6x - 3 + 5 = 2x + 10 | 1 |
Combine like terms: 6x + 2 = 2x + 10 | 1 |
Subtract 2x from both sides: 4x + 2 = 10 | 1 |
Subtract 2 from both sides: 4x = 8 | 1 |
Divide both sides by 4: | 1 |
Total | 5 |
Keep practicing, and you'll be a pro at isolating variables in no time! You've got this! ๐ช

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Question 1 of 9
What is the value of in the equation ?
4
10
21
14