Solve 7y(?-?)=-14xy+21: Finding Missing Terms in Linear Equations

Q: Why do I need to factor the right side first?

Factoring the right side reveals the structure you need to match! Without factoring $ -14xy + 21 $ into $ 7(-2xy + 3) $, you can't see what terms belong inside the parentheses.

Q: How do I handle the y that's outside the parentheses?

The y outside affects what goes inside the parentheses! Since we have $ 7y(? - ?) $, one missing term must have y in the denominator to cancel out: $ \frac{3}{y} $.

Q: What if I get the order of terms wrong?

Order matters for subtraction! The expression $ 7(-2xy + 3) $ becomes $ 7y(\frac{3}{y} - 2x) $, so first term is $ \frac{3}{y} $ and second term is $ 2x $.

Q: How can I check if my answer is correct?

Expand your answer and see if it matches! Take $ 7y(\frac{3}{y} - 2x) $ and distribute: $ 7y \cdot \frac{3}{y} - 7y \cdot 2x = 21 - 14xy $, which equals the original right side.

Factoring Equations with Missing Terms

Fill in the missing values:

$7y(?-?)=-14xy+21$

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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 Factor 14 into factors 7 and 2

00:11 Multiply by the appropriate whole fraction

00:17 Factor 21 into factors 7 and 3

00:21 Mark the common factors

00:27 Take out the common factors from the parentheses

00:51 Substitute to get subtraction

01:00 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:

$7y(?-?)=-14xy+21$

Step-by-step solution

To solve this problem, we need to factor both sides of the equation $7y(?-?) = -14xy + 21$ .

First, observe the terms on the right-hand side: $-14xy$ and $21$ .

Both terms share a common factor of 7. Factoring out 7 from both terms yields:

$7(-2xy + 3)$ .

Thus, the expression becomes:

$7y(-2x + \frac{3}{y})$ since the y was outside the parenthesis in the left.

Matching terms with $7y(? - ?)$ , the missing values are $\frac{3}{y}$ and $2x$ .

Therefore, the solution is: $\frac{3}{y}, 2x$ , corresponding to the correct choice.

Final Answer

$\frac{3}{y},2x$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find common factors first, then distribute carefully
Technique: Factor out 7 from -14xy + 21 to get 7(-2xy + 3)
Check: Expand $7y(\frac{3}{y} - 2x)$ to verify it equals -14xy + 21 ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing factors when matching terms
Don't assume the y outside parentheses stays there when rearranging = wrong term placement! This leads to incorrect identification of missing values. Always carefully distribute and match each term position exactly.

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 do I need to factor the right side first?

Factoring the right side reveals the structure you need to match! Without factoring $-14xy + 21$ into $7(-2xy + 3)$ , you can't see what terms belong inside the parentheses.

How do I handle the y that's outside the parentheses?

The y outside affects what goes inside the parentheses! Since we have $7y(? - ?)$ , one missing term must have y in the denominator to cancel out: $\frac{3}{y}$ .

What if I get the order of terms wrong?

Order matters for subtraction! The expression $7(-2xy + 3)$ becomes $7y(\frac{3}{y} - 2x)$ , so first term is $\frac{3}{y}$ and second term is $2x$ .

How can I check if my answer is correct?

Expand your answer and see if it matches! Take $7y(\frac{3}{y} - 2x)$ and distribute: $7y \cdot \frac{3}{y} - 7y \cdot 2x = 21 - 14xy$ , which equals the original right side.

Why is the answer not just simple integers?

Not all factoring problems have integer solutions! The term $\frac{3}{y}$ is needed because when multiplied by $7y$ , it gives us the constant term 21 on the right side.

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