Factoring a Variable from Multiple Terms to Isolate It
Grade 9 students in California Reveal Math Algebra 1 learn to isolate a variable that appears in multiple terms by factoring it out first. When the target variable appears in two or more terms, collect them on one side and factor: ap+bp=p(a+b). Then divide both sides by the entire grouped expression. For example, solving 4p-pq=12 for p: factor to p(4-q)=12, divide by (4-q) to get p=12/(4-q) where q≠4. A domain restriction always applies: set the denominator to zero to find which values must be excluded.
Key Concepts
When the variable you want to isolate appears in more than one term , factor it out first, then divide both sides by the remaining expression.
If $p$ appears in two terms, rewrite as: $$ap + bp = p(a + b)$$.
Common Questions
When do you need to factor a variable out of multiple terms?
When the variable you want to isolate appears in more than one term. You cannot use simple inverse operations alone — you must factor the variable out so it appears only once as a multiplied factor.
How do you solve 4p-pq=12 for p?
Factor p from both terms on the left: p(4-q)=12. Divide both sides by (4-q): p=12/(4-q). The domain restriction is q≠4 since the denominator cannot equal zero.
How do you solve rt+rs=d for r?
Factor r from both left terms: r(t+s)=d. Divide by (t+s): r=d/(t+s). Domain restriction: t≠-s.
What is a domain restriction and why does it matter?
A domain restriction is a value of a remaining variable that would make the denominator zero — which is undefined. Always check for this and state which values must be avoided.
What is the most common error when factoring a variable from multiple terms?
Dividing by only one part of the expression instead of the whole quantity. For p(4-q)=12, you must divide by (4-q), not just by 4.
Which unit covers this algebra technique in Algebra 1?
This skill is from Unit 1: Using Expressions and Equations in California Reveal Math Algebra 1, Grade 9.