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Chapter 1: Problem 32

Solve the given equation for \(x\). $$ \mathrm{x}+11=14 $$

Short Answer

Expert verified
The solution is \( x = 3 \).

Step by step solution

01

Understanding the Equation

The given equation is \( x + 11 = 14 \). Our goal is to find the value of \( x \) that makes this equation true.
02

Isolating x

To isolate \( x \) on one side, we need to get rid of the 11 on the left side of the equation. We can do this by subtracting 11 from both sides of the equation.
03

Performing the Calculation

Subtract 11 from both sides:\[x + 11 - 11 = 14 - 11\]This simplifies to:\[x = 3\]
04

Verification

Substitute \( x = 3 \) back into the original equation to ensure it's correct:\[x + 11 = 14\]Substitute \( x \):\[3 + 11 = 14\]Simplifying the right side, we find \( 14 = 14 \), confirming our solution is correct.

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Key Concepts

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

Equation Simplification
In solving linear equations like the given problem \(x + 11 = 14\), equation simplification is a crucial first step. This involves cleaning up the equation by performing operations that make it more straightforward without changing its meaning.
  • Start by identifying components of the equation you can simplify.
  • Choose operations that keep the equation balanced, such as adding or subtracting the same number on both sides.
For the equation \(x + 11 = 14\), we notice that simply subtracting 11 from both sides will help in simplifying it. This easy step transforms the equation into \(x = 3\), a much simpler form. Simplification sets us up perfectly to isolate the variable later on.
Isolation of Variable
Isolation of the variable is central to solving equations. Our target is to have the variable \(x\) all by itself on one side of the equation. This tells us exactly what value \(x\) must be. To do this effectively:
  • Look for constants or coefficients attached to \(x\) and think of operations to move them to the other side.
  • Use inverse operations, like subtraction to counter addition, or division to counter multiplication.
In our equation \(x + 11 = 14\), we achieved this by subtracting 11 from both sides. This operation helps in isolating \(x\) and leads to the simplified equation \(x = 3\). Isolation makes it easy to see the solution without additional steps.
Verification of Solution
Once a solution is suggested, verifying it ensures accuracy. This step checks if the original equation holds true when the determined value is plugged back in. Verification involves:
  • Substituting the solution back into the original equation.
  • Performing calculations to check if both sides equal each other.
In this case, substituting \(x = 3\) back into \(x + 11 = 14\) involves calculating \(3 + 11 = 14\). Since both sides match, the solution is verified as correct. Verification is the final assurance that the solution is accurate and reliable.

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