Distributing Before Collecting Variables
When solving equations with variables on both sides that contain parentheses, you must apply the distributive property first before collecting like terms — a fundamental step in enVision Algebra 1 Chapter 1 for Grade 11. For 3(x + 2) = 2x - 1, distribute to get 3x + 6 = 2x - 1, then collect variables to get x = -7. For 2(y - 4) = 5(y + 1), distribute both sides to get 2y - 8 = 5y + 5, then -3y = 13, y = -13/3. Some cases after distributing reveal no solution (variables cancel with unequal constants) or infinite solutions (variables cancel with equal constants).
Key Concepts
When solving equations with variables on both sides that contain parentheses, apply the distributive property first: $a(b + c) = ab + ac$. This eliminates parentheses before collecting like terms and variables.
Common Questions
Why must you distribute before collecting like terms?
The distributive property a(b+c) = ab + ac must remove the parentheses first. You cannot add x terms that are still multiplied by a coefficient inside parentheses.
How do you solve 3(x + 2) = 2x - 1?
Distribute: 3x + 6 = 2x - 1. Subtract 2x from both sides: x + 6 = -1. Subtract 6: x = -7.
How do you solve 2(y - 4) = 5(y + 1)?
Distribute both sides: 2y - 8 = 5y + 5. Subtract 2y: -8 = 3y + 5. Subtract 5: -13 = 3y. So y = -13/3.
What happens in 4(2x + 3) = 3(x - 1) + 5x?
Distribute: 8x + 12 = 3x - 3 + 5x = 8x - 3. Subtract 8x: 12 = -3. This is a contradiction — no solution.
When does distributing lead to infinite solutions?
When distributing and simplifying produces a statement like 5 = 5 (always true). This means the original equation is an identity and every value of the variable is a solution.