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Chapter 2: Problem 115

Solve each equation. Be sure to check each answer. $$-18+y=18$$

Short Answer

Expert verified
The solution to the equation \(-18 + y = 18\) is \( y = 36 \).

Step by step solution

01

Simplify and isolate the variable

To solve for \( y \), you will isolate \( y \) on one side of the equation. You can do this by adding \( 18 \) on both sides of the given equation to cancel out the negative on the left side. Doing this will give: \[-18 + 18 + y = 18 + 18\]. This simplifies to \( y = 36 \)
02

Verify the result

The final step is to verify the result by substituting \( y = 36 \) into the initial equation. To do this, plug in \( 36 \) for \( y \) in the equation \[-18 + y = 18\]. This gives \[-18 + 36 = 18\] which simplifies to \( 18 = 18 \). Since the equation is true, the solution \( y = 36 \) is correct.

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Key Concepts

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

Isolating the Variable
When solving a linear equation, like \(-18 + y = 18\), our primary goal is to isolate the variable. This means getting \(y\) by itself on one side of the equation. To do this here, we need to eliminate the \(-18\) that is currently added to \(y\). How can we achieve this? Simple: by using the inverse operation.

  • First, recognize that the opposite of \(-18\) is \(+18\).
  • Add \(+18\) to both sides of the equation: \(-18 + y + 18 = 18 + 18\).
  • On the left-hand side, \(-18 + 18\) cancels out, leaving \(y\) alone.
  • The right-hand side simplifies to \(36\).
By following these steps, we have isolated \(y\) and deduced that \(y = 36\). Remember, the core aim of isolating the variable is to simplify the equation so that one side contains only the variable, which then reveals its value.
Verification of Solutions
After isolating and finding \(y = 36\), it's essential to verify that our solution is correct. Verification is simply substituting your answer back into the original equation to make sure it satisfies the equation. By doing this, you confirm the accuracy of your work and ensure the proper solution.

  • Take the original equation: \(-18 + y = 18\).
  • Substitute \(36\) for \(y\): \(-18 + 36 = 18\).
  • Compute the left-hand side: \(-18 + 36\) results in \(18\).
Both sides of the equation are equal, thus confirming that \(18 = 18\). This tells us that our solution \(y = 36\) was indeed correct. Verifying solutions is a crucial step that validates your result and gives you confidence in your math skills.
Algebraic Simplification
Algebraic simplification is a fundamental process used throughout solving equations, including linear ones. It involves combining like terms and reducing the equation to its simplest form, making it easier to solve for the variable. Though in our exercise there wasn鈥檛 much complex simplification, it鈥檚 still key to understand the general process.

  • Combining like terms: When possible, gather terms that have the same variable or constant.
  • Use inverse operations: This involves adding, subtracting, multiplying, or dividing to remove terms and simplify expressions.
  • Coordinate balancing: Maintain equality by performing the same operation on both sides of an equation.
In the equation \(-18 + y = 18\), we mainly used inverse operations to simplify. By adding \(+18\) to both sides, the equation was reduced to a simple statement about \(y\). Simplification helps streamline your work and sets a clear path to solve equations with ease.

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Most popular questions from this chapter

Solve each inequality and graph the solution on the number line. $$-3 \leq \frac{x}{2} \leq 5$$

Discuss whether \(\frac{3}{2} x\) and \(\frac{3 x}{2}\) are like terms.

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To solve the inequality \(1 < \frac{1}{x},\) one student multiplies both sides by \(x\) to get \(x < 1 .\) Why is this not correct?

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