Expand (x/y - 3)(y + 1/4): Binomial Multiplication with Fractions

Q: Why does x/y times y equal just x?

When you multiply $ \frac{x}{y} \cdot y $, the y's cancel out! Think of it as $ \frac{x}{y} \cdot \frac{y}{1} = \frac{xy}{y} = x $.

Q: Do I need to combine like terms in this problem?

No! All four terms are different: x, $ \frac{x}{4y} $, -3y, and $ -\frac{3}{4} $. None can be combined since they have different variables or forms.

Q: How do I multiply fractions with variables?

Treat variables like numbers! For $ \frac{x}{y} \cdot \frac{1}{4} $, multiply numerators together and denominators together: $ \frac{x \cdot 1}{y \cdot 4} = \frac{x}{4y} $.

Q: What's the FOIL method for this problem?

First: $ \frac{x}{y} \cdot y = x $Outer: $ \frac{x}{y} \cdot \frac{1}{4} = \frac{x}{4y} $Inner: $ -3 \cdot y = -3y $Last: $ -3 \cdot \frac{1}{4} = -\frac{3}{4} $

Q: Why isn't the answer simplified further?

The expression $ x+\frac{x}{4y}-3y-\frac{3}{4} $ is already fully simplified! These are all different types of terms that cannot be combined or reduced further.

Binomial Expansion with Mixed Fractions

$(\frac{x}{y}-3)(y+\frac{1}{4})=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Open parentheses properly, multiply each factor by each factor

00:25 Calculate the multiplications

00:47 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

$(\frac{x}{y}-3)(y+\frac{1}{4})=$

Step-by-step solution

To solve the problem, we will use the distributive property to expand and simplify the expression $(\frac{x}{y}-3)(y+\frac{1}{4})$ .

Step 1: Apply the distributive property to each term in the first expression with each term in the second.

The expression $(\frac{x}{y})(y + \frac{1}{4}) - 3(y + \frac{1}{4})$ needs to be expanded:

Step 2: Distribute $\frac{x}{y}$ across $y$ and $\frac{1}{4}$ .

$\frac{x}{y} \cdot y = x.$
$\frac{x}{y} \cdot \frac{1}{4} = \frac{x}{4y}.$

Step 3: Distribute $-3$ across $y$ and $\frac{1}{4}$ .

$-3 \cdot y = -3y.$
$-3 \cdot \frac{1}{4} = -\frac{3}{4}.$

Step 4: Combine all distributed terms.

The complete expanded expression becomes:

$x + \frac{x}{4y} - 3y - \frac{3}{4}.$

Therefore, the solution to the problem is $\boxed{x+\frac{x}{4y}-3y-\frac{3}{4}}$ .

Final Answer

$x+\frac{x}{4y}-3y-\frac{3}{4}$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Multiply each term in first parentheses by each in second
Technique: $\frac{x}{y} \cdot y = x$ and $-3 \cdot \frac{1}{4} = -\frac{3}{4}$
Check: Verify four terms are present: two with x, one with y, one constant ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs
Don't just multiply -3 by y to get -3y and forget the fraction = missing -3/4 term! The negative must be applied to both terms inside the second parentheses. Always distribute the negative sign to every term in the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

Why does x/y times y equal just x?

When you multiply $\frac{x}{y} \cdot y$ , the y's cancel out! Think of it as $\frac{x}{y} \cdot \frac{y}{1} = \frac{xy}{y} = x$ .

Do I need to combine like terms in this problem?

No! All four terms are different: x, $\frac{x}{4y}$ , -3y, and $-\frac{3}{4}$ . None can be combined since they have different variables or forms.

How do I multiply fractions with variables?

Treat variables like numbers! For $\frac{x}{y} \cdot \frac{1}{4}$ , multiply numerators together and denominators together: $\frac{x \cdot 1}{y \cdot 4} = \frac{x}{4y}$ .

What's the FOIL method for this problem?

First: $\frac{x}{y} \cdot y = x$
Outer: $\frac{x}{y} \cdot \frac{1}{4} = \frac{x}{4y}$
Inner: $-3 \cdot y = -3y$
Last: $-3 \cdot \frac{1}{4} = -\frac{3}{4}$

Why isn't the answer simplified further?

The expression $x+\frac{x}{4y}-3y-\frac{3}{4}$ is already fully simplified! These are all different types of terms that cannot be combined or reduced further.

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