Verify Equation Balance: 26 = x/6 ?= 2/x = 1/78

Q: What does the question mark symbol mean in the equation?

The question mark means we need to verify if that part of the equation is true. It's asking: "Does $ \frac{x}{6} $ actually equal $ \frac{2}{x} $?"

Q: Why do I start with the equation that doesn't have a question mark?

Start with $ \frac{2}{x} = \frac{1}{78} $ because this is a known equality. This gives you the value of x, which you can then use to check all other parts.

Q: How do I solve 2/x = 1/78 for x?

Cross-multiply: $ 2 \times 78 = x \times 1 $, so $ x = 156 $. Or multiply both sides by x, then by 78.

Q: What if I got a different value of x?

Double-check your cross-multiplication! From $ \frac{2}{x} = \frac{1}{78} $, you should get $ 2 \times 78 = 156 $. Then verify: $ \frac{2}{156} = \frac{1}{78} $ ✓

Q: Do all parts of the equation chain have to equal exactly the same number?

Yes! For the equations to be "balanced," every single part must equal the same value. In this case, everything must equal 26.

Equation Balance with Rational Expressions

Are the equations balanced?

$26{=}\frac{x}{6}\stackrel{?}{=}\frac{2}{x}{=}\frac{1}{78}$

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Understand the problem

Are the equations balanced?

$26{=}\frac{x}{6}\stackrel{?}{=}\frac{2}{x}{=}\frac{1}{78}$

Step-by-step solution

To determine whether the given equations are balanced, we begin by examining them step-by-step:

Given: $26 = \frac{x}{6} \stackrel{?}{=} \frac{2}{x} = \frac{1}{78}$

We should first determine the value of $x$ from the equation involving $\frac{1}{78}$ :

The last equation is $\frac{1}{78} = \frac{2}{x}$ .
Multiplying both sides by $x$ and then by $78$ gives $x = 156$ .

Use $x = 156$ to evaluate the other expressions:

For $\frac{x}{6}$ :
$\frac{156}{6} = 26$

Now, check if all values equate:

$26 = 26$ from $\frac{x}{6}$
$26 = 26$ from the right side of the original problem statement

All values match, indicating:

The equations are balanced. Therefore, the correct answer is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Balance Check: All expressions must equal the same value
Technique: Solve $\frac{2}{x} = \frac{1}{78}$ to find x = 156
Verify: Substitute x = 156: $\frac{156}{6} = 26$ and 26 = 26 ✓

Common Mistakes

Avoid these frequent errors

Checking only adjacent equations instead of all parts
Don't just verify that $\frac{x}{6} = \frac{2}{x}$ without checking the entire chain! This misses whether the question mark equation actually holds. Always solve for x from the known equality and verify every single part of the chain equals the same value.

Practice Quiz

Test your knowledge with interactive questions

Are the equations balanced?

$9-x=1\stackrel{?}{=}5x=-40$

FAQ

Everything you need to know about this question

What does the question mark symbol mean in the equation?

The question mark means we need to verify if that part of the equation is true. It's asking: "Does $\frac{x}{6}$ actually equal $\frac{2}{x}$ ?"

Why do I start with the equation that doesn't have a question mark?

Start with $\frac{2}{x} = \frac{1}{78}$ because this is a known equality. This gives you the value of x, which you can then use to check all other parts.

How do I solve 2/x = 1/78 for x?

Cross-multiply: $2 \times 78 = x \times 1$ , so $x = 156$ . Or multiply both sides by x, then by 78.

What if I got a different value of x?

Double-check your cross-multiplication! From $\frac{2}{x} = \frac{1}{78}$ , you should get $2 \times 78 = 156$ . Then verify: $\frac{2}{156} = \frac{1}{78}$ ✓

Do all parts of the equation chain have to equal exactly the same number?

Yes! For the equations to be "balanced," every single part must equal the same value. In this case, everything must equal 26.

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