Answered

Choose Yes or No to tell if you can use the equation to solve the problem below.



$?-23=61$



YesNo

$61-?=23$		(A)(B)

$23+61=?$		(C)(D)

$61-23=?$		(E)(F)

$?-61=23$		(G)(H)

Verified Answer :

  • The initial equation is \(x - 23 = 61\), which solves to \(x = 84\).
  • Checking each option for equivalence:
  • \(61 - x = 23 \implies x = 38\) (No)
  • \(23 + 61 = x \implies x = 84\) (Yes)
  • \(61 - 23 = x \implies x = 38\) (No)
  • \(x - 61 = 23 \implies x = 84\) (Yes)
  • Final Results: \(\boxed{\text{No, Yes, No, Yes}}\)

Explanation

  1. Analyzing the Initial Equation and Solving for the Unknown

Let the unknown value be represented by \(x\). The given initial equation is \(x - 23 = 61\). To find the value of \(x\), we add 23 to both sides of the equation:
\[x = 61 + 23 = 84.\]
Now, we must evaluate each of the four proposed equations to determine if they are equivalent to the original equation, meaning they must also yield the result \(x = 84\).

  1. Evaluating the Proposed Equations

We evaluate each of the four provided options one by one:

  1. For the equation \(61 - x = 23\):

Subtract 61 from both sides: \(-x = 23 - 61\), so \(-x = -38\), which means \(x = 38\). This does not match 84. (No)

  1. For the equation \(23 + 61 = x\):

Calculating the right side: \(23 + 61 = 84\), so \(x = 84\). This matches our solution. (Yes)

  1. For the equation \(61 - 23 = x\):

Calculating the right side: \(61 - 23 = 38\), so \(x = 38\). This does not match 84. (No)

  1. For the equation \(x - 61 = 23\):

Add 61 to both sides: \(x = 23 + 61 = 84\). This matches our solution. (Yes)

  1. Final Conclusion

Based on the evaluations, we can summarize the results for each equation:

  • The equation \(61 - ? = 23\) does not solve for 84, so the answer is No.
  • The equation \(23 + 61 = ?\) solves for 84, so the answer is Yes.
  • The equation \(61 - 23 = ?\) does not solve for 84, so the answer is No.
  • The equation \(? - 61 = 23\) solves for 84, so the answer is Yes.

Thus, the final results are:

  • \(61 - ? = 23\) (No)
  • \(23 + 61 = ?\) (Yes)
  • \(61 - 23 = ?\) (No)
  • \(? - 61 = 23\) (Yes)

Examples

Understanding how to rearrange equations is essential in real-world problem solving. For example, if you know that the total cost of a repair is \$61, and you've already paid a deposit of \$23, you can model the remaining amount as \(x - 23 = 61\), which helps you realize you still need \$84 to settle the bill. Being able to recognize equivalent forms of equations, such as \(x = 61 + 23\), allows you to quickly calculate the missing values in various daily financial or scheduling scenarios.

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