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Chapter 2: Problem 14

Solve each equation using the Subtraction and Addition Properties of Equality. $$a-30=57$$

Short Answer

Expert verified
a = 87

Step by step solution

01

Identify the Equation

The given equation is: \[ a - 30 = 57 \]
02

Isolate the Variable

To isolate the variable \(a\), add 30 to both sides of the equation to cancel out the subtraction on the left side: \[ a - 30 + 30 = 57 + 30 \]
03

Simplify Both Sides

Simplify both sides of the equation: \[ a = 87 \]

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Key Concepts

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

Subtraction Property of Equality
The Subtraction Property of Equality is a foundational concept in algebra. It states that if you subtract the same number from both sides of an equation, the two sides remain equal. This property helps in simplifying and solving equations. For example, in the equation \( a - 30 = 57 \), we need to move the constant term (\(-30\)) to isolate the variable. Subtraction helps balance the equation, ensuring it remains valid.
Using this property correctly is crucial, as it helps keep the equation balanced while simplifying. Remember to perform the same subtraction on both sides to maintain the equation鈥檚 equality.
Addition Property of Equality
The Addition Property of Equality is similar to the subtraction property, but focuses on adding the same number to both sides of an equation. This ensures that the equation stays balanced. In the context of our equation \(a - 30 = 57\), we use this property to eliminate the \(-30\).
Here鈥檚 how it鈥檚 done: adding 30 to both sides of the equation. \[ a - 30 + 30 = 57 + 30 \]. By doing this, we cancel out the \(-30\) on the left side, leaving us with \(a\):
\[ a = 87 \]. This demonstrates how addition helps isolate the variable, making the equation simpler and easier to solve.
Solving Equations
Solving equations is a key skill in elementary algebra. It involves finding the value of the variable that makes the equation true. Here鈥檚 a step-by-step breakdown:
  • Identify the equation: Spot the variable and constants.
  • Isolate the variable: Use addition or subtraction properties to move constants from the variable鈥檚 side.
  • Simplify both sides: After isolating the variable, perform calculations to simplify the equation.

In our example, given \(a - 30 = 57\):
  • We identify the equation and the need to isolate \(a\).
  • We add 30 to both sides using the Addition Property of Equality: \[a - 30 + 30 = 57 + 30.\]
  • Simplify to find \[a = 87.\]

This process is systematic and helps solve equations efficiently.

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