Apply the distributive property to factor out the greatest common factor of all three terms.
$9-12 x+6 y=$
Answer (2)
Find the greatest common factor (GCF) of the coefficients: GCF(9, -12, 6) = 3.
Rewrite each term as a product of the GCF and another factor: 9 = 3 × 3 , − 12 x = 3 × − 4 x , 6 y = 3 × 2 y .
Apply the distributive property to factor out the GCF: 9 − 12 x + 6 y = 3 ( 3 − 4 x + 2 y ) .
The factored expression is 3 ( 3 − 4 x + 2 y ) .
Explanation
Understanding the Problem We are asked to factor out the greatest common factor (GCF) from the expression 9 − 12 x + 6 y . This involves identifying the largest number that divides evenly into all the coefficients (9, -12, and 6) and then using the distributive property to rewrite the expression.
Finding the GCF First, we need to find the greatest common factor (GCF) of the coefficients 9, -12, and 6. The factors of 9 are 1, 3, and 9. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 6 are 1, 2, 3, and 6. The greatest common factor of 9, 12, and 6 is 3.
Rewriting the Terms Now, we rewrite each term in the expression as a product of the GCF (which is 3) and another factor:
9 = 3 × 3
− 12 x = 3 × ( − 4 x )
6 y = 3 × 2 y
Factoring Out the GCF Using the distributive property, we factor out the GCF (3) from the expression:
9 − 12 x + 6 y = 3 ( 3 ) + 3 ( − 4 x ) + 3 ( 2 y ) = 3 ( 3 − 4 x + 2 y )
Final Answer Therefore, the expression 9 − 12 x + 6 y factored using the distributive property and the greatest common factor is 3 ( 3 − 4 x + 2 y ) .
Examples
Factoring out the greatest common factor is useful in simplifying algebraic expressions and solving equations. For example, if you have an equation like 9 − 12 x + 6 y = 0 , factoring out the GCF allows you to rewrite it as 3 ( 3 − 4 x + 2 y ) = 0 . This can make it easier to analyze the equation and find solutions, especially when dealing with more complex expressions or equations in fields like physics or engineering where simplifying expressions can make calculations more manageable.
To factor the expression 9 − 12 x + 6 y , we find the greatest common factor (GCF) of its coefficients, which is 3. We can then rewrite each term as a product of the GCF and factor it out, resulting in 3 ( 3 − 4 x + 2 y ) . This method helps simplify the expression and can be useful in solving related problems.
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