Solve the Equation: 3y(?-?) = 21xy + 9 | Finding Missing Terms

Q: Why is the second term negative 3/y instead of positive?

When we have $ 21xy + 9 $ and factor out 3, we get $ 3(7xy + 3) $. To match $ 3y(? - ?) $, we need subtraction so that $ 3y \cdot \frac{-3}{y} = -9 $, not +9.

Q: How do I know what to factor out first?

Look for the greatest common factor (GCF) of all terms. In $ 21xy + 9 $, both terms are divisible by 3, so factor out 3 first: $ 3(7xy + 3) $.

Q: Why can't the answer be 18x and 3?

If we use 18x and 3, then $ 3y(18x - 3) = 54xy - 9y $, which doesn't equal $ 21xy + 9 $. Always check your work by expanding!

Q: What does the fraction 3/y mean in this context?

The fraction $ \frac{3}{y} $ is a rational expression. When multiplied by y, it gives us 3: $ y \cdot \frac{3}{y} = 3 $. This helps balance our equation.

Q: How can I double-check my factoring?

Always expand your answer and see if it matches the original expression. Multiply $ 3y(7x - \frac{3}{y}) $ to get $ 21xy - 9 $, which should equal the given expression.

Algebraic Factorization with Mixed Terms

Fill in the missing values:

$3y(?-?)=21xy+9$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing values

00:04 Factorize 21 into factors 3 and 7

00:10 Factorize 9 into factors 3 and 3

00:15 Multiply by the appropriate whole fraction

00:30 Mark the common factors

00:43 Take out the common factors from the parentheses

00:50 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing values:

$3y(?-?)=21xy+9$

Step-by-step solution

To solve the problem, follow these steps:

Step 1: Recognize the expression $21xy + 9$ needs to be matched by factoring it as a common product expression that includes $3y$ .
Step 2: Identify the greatest common factor in $21xy + 9$ , which is $3$ . Thus, the factorization is $3(7xy + 3)$ .
Step 3: Now, we need $3y(?-?) = 3(7xy + 3)$ . Since we factor out a $3$ , the matching terms should sum up to $y(7x) + y\left(\frac{-3}{y}\right)$ .
Step 4: Match the missing numbers found in the expression: $? - ? = 7x, \frac{-3}{y}$ .

By matching, the factors yield $3y(?-?) = 3y(7x - \frac{3}{y})$ . This confirms the missing values are $7x$ and $\frac{-3}{y}$ .

Therefore, the correct completion of the expression is $7x, \frac{-3}{y}$ , which corresponds to choice 4.

Final Answer

$7x,\frac{-3}{y}$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Factor out greatest common factor first from expression
Technique: From 21xy + 9, factor out 3 to get 3(7xy + 3)
Check: Multiply 3y(7x - 3/y) = 21xy - 9 to verify ✓

Common Mistakes

Avoid these frequent errors

Not properly distributing the factored terms
Don't forget that 3y(7x - 3/y) requires distributing 3y to both terms = 21xy - 9! Students often only multiply the first term. Always distribute the outside factor to every term inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why is the second term negative 3/y instead of positive?

When we have $21xy + 9$ and factor out 3, we get $3(7xy + 3)$ . To match $3y(? - ?)$ , we need subtraction so that $3y \cdot \frac{-3}{y} = -9$ , not +9.

How do I know what to factor out first?

Look for the greatest common factor (GCF) of all terms. In $21xy + 9$ , both terms are divisible by 3, so factor out 3 first: $3(7xy + 3)$ .

Why can't the answer be 18x and 3?

If we use 18x and 3, then $3y(18x - 3) = 54xy - 9y$ , which doesn't equal $21xy + 9$ . Always check your work by expanding!

What does the fraction 3/y mean in this context?

The fraction $\frac{3}{y}$ is a rational expression. When multiplied by y, it gives us 3: $y \cdot \frac{3}{y} = 3$ . This helps balance our equation.

How can I double-check my factoring?

Always expand your answer and see if it matches the original expression. Multiply $3y(7x - \frac{3}{y})$ to get $21xy - 9$ , which should equal the given expression.

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