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Chapter 2: Problem 110

Solve each equation. Be sure to check each answer. $$-93=67+y$$

Short Answer

Expert verified
The solution to the equation is \(y = -160\).

Step by step solution

01

Analyzing the equation

The given equation is \(-93 = 67+y\). The goal is to isolate 'y' on one side of the equation. To do that, subtract 67 from both sides.
02

Subtract 67 from both sides

Subtracting 67 from each side of the equation, we get: \[ -93 -67 = 67 -67 + y \]. Upon subtraction, this simplifies to \[-160 = y\].
03

Checking the answer

Check the answer by substituting the solution back into the original equation. If the left-hand side equals the right-hand side, then the solution is correct.\[ -93 = 67 -160 \]. After simplifying the right-hand side, \[ -93 = -93 \], which confirms the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Isolation
In algebra, isolating a variable is an essential skill. This means rearranging an equation so that the variable you are solving for stands alone on one side. To do this with the given equation \(-93 = 67 + y\), our aim is to have \(y\) all by itself. This involves performing operations to both sides of the equation equally.
  • Identify the terms that need to be moved. Here, it鈥檚 the \(67\) on the same side as \(y\).
  • Decide on the operation needed to cancel the unwanted term. Since \(67\) is added to \(y\), subtraction is the logical choice.
Remember, whatever operation you perform on one side must also be done on the other side. This keeps the equation balanced, like a perfectly adjusted scale.
Integer Subtraction
Integer subtraction is a fundamental arithmetic operation, and it plays a critical role in solving linear equations. In our equation, we subtracted 67 from each side. Let鈥檚 break down the process.
  • We started with \(-93 - 67\). This is the same as adding a negative number, \(-67\), to \(-93\).
  • When subtracting integers, align them by their signs: \(-93\) and \(-67\) are both negative, so their absolute values add up as if we are adding positives: \(93 + 67 = 160\).
  • However, since both numbers are negative, the result remains negative: \(-160\).
Understanding that subtraction can be a form of addition helps, because it鈥檚 all about combining numbers correctly based on their signs.
Solution Verification
Verifying a solution is like double-checking your work to ensure it鈥檚 correct. After solving \(y = -160\), it's crucial to plug this back into the original equation to confirm it satisfies the equation. For verification:
  • Substitute \(-160\) for \(y\) in the equation: \(-93 = 67 + (-160)\).
  • Simplify the equation. Calculate \(67 - 160\) to get \(-93\).
  • Compare both sides: if they鈥檙e equal, your solution is verified as correct.
Verification ensures reliability and accuracy in solving equations. It鈥檚 a simple, yet essential, check to validate your solution and strengthen your understanding.

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