Simplify 3x-(y·z+3z:z/x): Multiple Variable Expression Solution

Q: What does the colon (:) mean in this expression?

The colon (:) means division! So $ 3z : \frac{z}{x} $ is the same as $ 3z \div \frac{z}{x} $. This notation is common in some textbooks.

Q: How do I divide by a fraction?

To divide by a fraction, multiply by its reciprocal! So $ 3z \div \frac{z}{x} = 3z \times \frac{x}{z} $. Flip the fraction and change division to multiplication.

Q: Why do the 3x terms cancel out?

Because we have $ 3x - 3x $ when we distribute the negative sign. When you subtract a number from itself, you always get zero! This leaves only $ -yz $.

Q: Should I simplify inside the parentheses first?

Yes! Always follow the order of operations (PEMDAS). Work inside parentheses first, then apply any operations outside. This prevents mistakes and keeps your work organized.

Q: What if I forgot to distribute the negative sign?

You'd get $ 3x - yz - 3x $ instead of $ 3x - yz - 3x $. The negative sign affects every term inside the parentheses. Practice writing it out step by step!

Order of Operations with Division Notation

$3x-(y\cdot z+3z:\frac{z}{x})=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Negative multiplied by positive is always negative

00:16 Division is also multiplication by the inverse

00:26 Let's reduce what we can

00:30 Let's group terms

00:33 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

$3x-(y\cdot z+3z:\frac{z}{x})=\text{?}$

Step-by-step solution

To solve this problem, we begin by simplifying the expression inside the parentheses: $y \cdot z + 3z : \frac{z}{x}$ .

First, evaluate the division: $\frac{3z}{\frac{z}{x}}$ . This can be simplified by multiplying by the reciprocal, yielding $3z \cdot \frac{x}{z} = 3x$ .

The expression inside the parentheses becomes $y \cdot z + 3x$ .

Now substitute this back into the entire expression: $3x - (y \cdot z + 3x)$ .

Apply the distributive property to the negative sign, resulting in: $3x - y \cdot z - 3x$ .

Combine like terms: $3x - 3x - y \cdot z$ simplifies to $- y \cdot z$ .

Thus, the solution to the given expression is $-yz$ .

Final Answer

$-yz$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Transform division by fraction into multiplication by reciprocal
Technique: $3z : \frac{z}{x} = 3z \cdot \frac{x}{z} = 3x$
Check: Verify by substituting: if $3x - (yz + 3x) = -yz$ ✓

Common Mistakes

Avoid these frequent errors

Treating division notation incorrectly
Don't interpret $3z : \frac{z}{x}$ as $\frac{3z}{z}{x}$ = wrong simplification! This misunderstands the colon notation and leads to incorrect terms. Always convert division by a fraction to multiplication by its reciprocal first.

Practice Quiz

Test your knowledge with interactive questions

$60:(10\times2)=$

FAQ

Everything you need to know about this question

What does the colon (:) mean in this expression?

The colon (:) means division! So $3z : \frac{z}{x}$ is the same as $3z \div \frac{z}{x}$ . This notation is common in some textbooks.

How do I divide by a fraction?

To divide by a fraction, multiply by its reciprocal! So $3z \div \frac{z}{x} = 3z \times \frac{x}{z}$ . Flip the fraction and change division to multiplication.

Why do the 3x terms cancel out?

Because we have $3x - 3x$ when we distribute the negative sign. When you subtract a number from itself, you always get zero! This leaves only $-yz$ .

Should I simplify inside the parentheses first?

Yes! Always follow the order of operations (PEMDAS). Work inside parentheses first, then apply any operations outside. This prevents mistakes and keeps your work organized.

What if I forgot to distribute the negative sign?

You'd get $3x - yz - 3x$ instead of $3x - yz - 3x$ . The negative sign affects every term inside the parentheses. Practice writing it out step by step!

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