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Chapter 2: Problem 70

Explain why \(x+2=9\) and \(x+2=-6\) are not equivalent equations.

Short Answer

Expert verified
The equations \(x+2=9\) and \(x+2=-6\) are not equivalent because their solutions, \(x=7\) for the first equation and \(x=-8\) for the second one, are not the same.

Step by step solution

01

Identify the Equations

The problem gives two linear equations: \(x+2=9\) and \(x+2=-6\). These are the two equations which we need to analyze.
02

Solve the First Equation

First, solve the equation \(x+2=9\) for 'x'. To do this, we subtract '2' from both sides of the equation, isolating 'x' on one side. Doing so gives us \(x=9-2\), so \(x=7\).
03

Solve the Second Equation

Next, solve the equation \(x+2=-6\) for 'x'. Again, we subtract '2' from both sides of the equation. This gives us \(x=-6-2\), so \(x=-8\).
04

Compare the Solutions

Now we compare the solutions. The solution to the first equation is \(x=7\), and the solution to the second equation is \(x=-8\). Because these solutions are different, the two original equations are not equivalent.

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Key Concepts

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

Solving Equations
Solving linear equations is an essential concept in algebra. In essence, it involves finding the value of the unknown variable that makes the equation true. When faced with an equation like \(x + 2 = 9\), your goal is to isolate \(x\) by performing operations that maintain the equality on both sides. This is essential because the operations applied must yield an equivalent equation.Here's a simple breakdown of solving the equation \(x + 2 = 9\):
  • To isolate \(x\), you need to undo the operations affecting it. Here, \(x\) is being added to 2, so you subtract 2.
  • Subtracting 2 from both sides gives \(x + 2 - 2 = 9 - 2\), simplifying to \(x = 7\).
  • Now, \(x = 7\) is your solution, as substituting \(x\) with 7 balances the equation.
Remember, the same principle applies when solving the second equation \(x + 2 = -6\), where the solution is \(x = -8\) after performing similar steps. The key is ensuring the same operation is applied to both sides of the equation.
Equivalent Equations
Equivalent equations are equations that have exactly the same solution or solutions. They might look different, but they represent the same relationship. In the case of linear equations, applying the same mathematical operation to both sides should result in an equivalent equation, allowing us to eventually find the unknown variable.In the exercise, the equations \(x + 2 = 9\) and \(x + 2 = -6\) are not equivalent. Here's why:
  • Solving the first gives \(x = 7\) as the solution.
  • The second equation's solution is \(x = -8\).
  • Since the solutions \(x = 7\) and \(x = -8\) are not the same, the equations are not equivalent.
Understanding equivalent equations helps you recognize when multiple equations represent the same problem or scenario, which is crucial in more complex algebraic contexts. They ensure consistency throughout operations.
Isolation of Variables
Isolation of variables is at the heart of solving linear equations. It refers to manipulating the equation to have the unknown variable on one side and the constants on the other. This process makes it clear what value the variable should take to satisfy the equation fully.Consider the equation \(x + 2 = 9\). Your goal is to have \(x\) isolated.
  • Begin by eliminating the constant attached to \(x\), here it's 2.
  • Subtract 2 from both sides to remove it, achieving \(x = 9 - 2\), thus \(x = 7\).
  • For \(x + 2 = -6\), perform the same step: subtract 2 to isolate \(x\), resulting in \(x = -8\).
By mastering the isolation of variables, you'll simplify solving not just linear equations, but will also prepare yourself for dealing with more advanced algebraic topics, ensuring you're capable of handling equations involving multiple steps or variables.

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