Verify: Are the Equations x-25=5 and 7x-10=200 Balanced?

Q: What does it mean for equations to be 'balanced'?

Two equations are balanced when they have the same solution. Even if they look completely different, if they both give the same x-value, they are balanced!

Q: Do I need to solve both equations completely?

Yes! You must find the exact value of x in each equation. Only then can you compare the solutions to see if they match.

Q: What if one equation has x = 30 and the other has x = -30?

Then the equations are not balanced because they have different solutions. The sign matters - positive and negative values are different!

Q: Can I just substitute x = 30 without solving first?

No! You need to solve each equation independently first. Then you can verify your work by substituting the solution back into both original equations.

Equation Verification with Common Solutions

Are the equations balanced?

$x-25=5\stackrel{?}{=}7x-10=200$

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Understand the problem

Are the equations balanced?

$x-25=5\stackrel{?}{=}7x-10=200$

Step-by-step solution

We will solve each equation step-by-step to ascertain if there is a common solution indicative of them being balanced:

Solving the first equation $x - 25 = 5$ :

Add 25 to both sides to isolate $x$ :

x - 25 + 25 = 5 + 25

x = 30

Solving the second equation $7x - 10 = 200$ :

Add 10 to both sides to isolate the term with $x$ :

7x - 10 + 10 = 200 + 10

7x = 210

Divide by 7 to solve for $x$ :

x = \frac{210}{7}

x = 30

Both equations give the solution $x = 30$ . This indicates that they are indeed balanced, as they share a common solution for $x$ .

Therefore, the equations are balanced.

The correct answer to the problem is: Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Balance Rule: Two equations are balanced if they share the same solution
Technique: Solve each equation separately: x - 25 = 5 gives x = 30
Check: Substitute x = 30 into both equations to verify balance ✓

Common Mistakes

Avoid these frequent errors

Assuming equations are balanced without solving
Don't just compare the equations visually = wrong conclusion! Different-looking equations can have the same solution. Always solve each equation completely to find their solutions, then compare the x-values.

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 it mean for equations to be 'balanced'?

Two equations are balanced when they have the same solution. Even if they look completely different, if they both give the same x-value, they are balanced!

Do I need to solve both equations completely?

Yes! You must find the exact value of x in each equation. Only then can you compare the solutions to see if they match.

What if one equation has x = 30 and the other has x = -30?

Then the equations are not balanced because they have different solutions. The sign matters - positive and negative values are different!

Can I just substitute x = 30 without solving first?

No! You need to solve each equation independently first. Then you can verify your work by substituting the solution back into both original equations.

What if both equations give me the same steps?

That's a good sign, but you still need to complete the algebra for each one. Sometimes equations look similar but have small differences that change the final answer.

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