Complete the Expression: 9a+8b+7c+?=9a+12b+7c

Q: Why can't the answer just be 4b?

Look carefully at the expression: we have bc being subtracted. If you put 4b in the blank, you get $ 4b - bc $, which doesn't simplify to 4b. You need $ \frac{4}{c} $ so that $ \frac{4}{c} \cdot c = 4 $.

Q: How can I check my answer?

Substitute $ \frac{4}{c} $ into the original expression: $ 9a + 8b + 7c + \frac{4}{c} - bc $. The $ \frac{4}{c} $ term doesn't change, but we still get the equivalent expression we need.

Q: What if c equals zero?

If c = 0, then $ \frac{4}{c} $ would be undefined. In algebra problems like this, we assume all variables represent non-zero values unless stated otherwise.

Algebraic Expressions with Fractional Terms

Fill in the blank:

^{$9a+8b+7c+_—bc=9a+12b+7c$}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 First, let's find the unknown value.

00:14 Next, we'll simplify as much as possible.

00:19 Now, let's isolate the unknown and solve for it.

00:29 That's how we find the solution to this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the blank:

^{$9a+8b+7c+_—bc=9a+12b+7c$}

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Compare both sides of the equation
Step 2: Identify and equate the coefficients of similar terms
Step 3: Solve for the missing term

Now, let's work through each step:
Step 1: The equation we need to balance is $9a + 8b + 7c + \_ - bc = 9a + 12b + 7c$ .
Step 2: Compare terms in both expressions. We see that $9a$ and $7c$ are matched on both sides. For the $b$ terms, we have $8b$ on the left and $12b$ on the right, indicating that the left-hand side needs an additional $4b$ to balance $b$ terms.
Step 3: The missing term must cancel $bc$ and add $4b$ . This means the term should be $\frac{4b}{c}$ because $\frac{4}{c} \cdot c = 4$ , thus giving us the required $4b$ .

Therefore, the solution to the problem is $\frac{4}{c}$ .

Final Answer

$\frac{4}{c}$

Key Points to Remember

Essential concepts to master this topic

Term Comparison: Match like terms on both sides to identify differences
Missing Value: Calculate $12b - 8b = 4b$ to find needed term
Verification: Substitute to check: $\frac{4}{c} \cdot c = 4$ gives required coefficient ✓

Common Mistakes

Avoid these frequent errors

Assuming the blank equals 4b directly
Don't write 4b in the blank = wrong dimensional analysis! The expression has bc being subtracted, so you need a term that when multiplied by c gives 4b. Always consider what happens when the term interacts with existing variables.

Practice Quiz

Test your knowledge with interactive questions

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

FAQ

Everything you need to know about this question

Why can't the answer just be 4b?

Look carefully at the expression: we have bc being subtracted. If you put 4b in the blank, you get $4b - bc$ , which doesn't simplify to 4b. You need $\frac{4}{c}$ so that $\frac{4}{c} \cdot c = 4$ .

How do I know what coefficient the b term needs?

Compare the b terms on both sides: left side has 8b and right side has 12b. The difference is $12b - 8b = 4b$ , so you need to add 4b to the left side.

Why is the answer a fraction?

Since we're subtracting bc in the expression, we need a term that when multiplied by c gives us +4b. That term is $\frac{4}{c}$ because $\frac{4}{c} \times c = 4$ .

How can I check my answer?

Substitute $\frac{4}{c}$ into the original expression: $9a + 8b + 7c + \frac{4}{c} - bc$ . The $\frac{4}{c}$ term doesn't change, but we still get the equivalent expression we need.

What if c equals zero?

If c = 0, then $\frac{4}{c}$ would be undefined. In algebra problems like this, we assume all variables represent non-zero values unless stated otherwise.

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