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Chapter 3: Problem 26

Use the properties of equality to help solve each equation. $$ 13-x=47 $$

Short Answer

Expert verified
The solution to the equation is \(x = -34\).

Step by step solution

01

Identify the Equation

The given equation is: \[13 - x = 47\]. Our goal is to isolate the variable \(x\).
02

Subtract Constant on Both Sides

Subtract 13 from both sides of the equation to begin isolating \(x\). This gives us: \[13 - x - 13 = 47 - 13\] which simplifies to: \[-x = 34\].
03

Multiply by -1 to Solve for x

To get \(x\) by itself, multiply both sides of the equation by -1 to change the sign of \(x\): \[-1(-x) = -1(34)\] which simplifies to: \[x = -34\].

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Key Concepts

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

Solving Equations
When approaching an equation, the ultimate goal is to find the value of the variable that makes the equation true. This requires a systematic approach. Begin with
  • Understanding the Equation: Recognize the equation format. Here, we have a basic linear equation, \[13 - x = 47\].
  • Perform Operations Step-by-Step: Begin solving by performing operations that simplify the equation while keeping it balanced.
The properties of equality, such as the subtraction property, are essential here. They allow us to manipulate the equation maintaining its truth by ensuring whatever operation is done on one side is also done on the other.
Solving equations involves multiple steps depending on complexity, but the fundamental strategy remains: Keep the equation balanced and reduce it towards finding the variable value.
Isolate the Variable
Isolating the variable means rearranging the equation so that the variable appears alone on one side. In our example, the task is to get\[-x\] by itself on one side:
  • By subtracting 13 from both sides, we simplify the equation to \[-x = 34\]. This isolates the variable term on the left side.
  • Notice that isolating the term \[-x\] is just halfway; eventually, you must solve for \[x\].
When isolating a variable, always ensure operations are mirrored on both sides to keep the equation balanced. This method paves the way to solving the equation completely. Once isolated, you can easily perform additional steps to reveal the variable's value.
Subtraction Property
The subtraction property of equality is a crucial rule that aids in solving equations. It states that subtracting the same number from both sides of an equation does not change the equation's truth.
  • In our case, subtracting 13 from both sides removes 13 from left, modifying the equation from \[13 - x = 47\] to \[-x = 34\].
  • This step is vital as it progresses towards isolating the variable.
Without the subtraction property, it would be challenging to simplify and solve complex equations. This property maintains the balance of an equation, enabling the breaking down of equations into simpler components that can be easily solved.
Remember, each step in the equation-solving process relies significantly on these fundamental properties being applied correctly and consistently.

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