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Chapter 5: Problem 72

Solve the equation. $$ 2 x-6=20 $$

Short Answer

Expert verified
The solution to the equation \(2x - 6 = 20\) is \(x = 13\).

Step by step solution

01

Apply addition inverse

First of all, apply the addition inverse to get rid of '-6' on the left side of equation. The addition inverse of '-6' is '+6'. So, add '6' to both sides of the equation: \[2x - 6 + 6 = 20 + 6\], which simplifies to \[2x = 26\].
02

Apply multiplication inverse

To keep 'x' alone in one side, divide both sides of the equation by '2', which is the Multiplicative Inverse of '2'. This gives: \[2x / 2 = 26 / 2\], which further simplifies to \[x = 13\].

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Key Concepts

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

Addition Inverse
In math, especially when solving equations, using the addition inverse is a handy technique. The addition inverse of a number is what you need to add to it, to get zero. This is essential when we want to isolate variables in linear equations.Let's take an example: If we have an equation like this: \(2x - 6 = 20\), we want to get rid of that \(-6\) on the left side. The addition inverse of \(-6\) is \(+6\).
  • Adding \(+6\) to \(-6\) gives us zero.
  • We also need to add \(+6\) to the other side to keep the equation balanced.
So, we modify our equation as follows: \(2x - 6 + 6 = 20 + 6\). This simplifies to \(2x = 26\).
This step is crucial because it simplifies the equation, making it easier to solve the unknown variable, "x" in this case.
Multiplicative Inverse
Once we've simplified the problem using addition inverse, we can now focus on isolating the variable completely. This is where the multiplicative inverse comes into play. The multiplicative inverse, often referred to as "reciprocal," of a number is what you multiply by to get one.For example, if we have \(2x = 26\), we want to get \( x\) alone.
  • The multiplicative inverse of \(2\) is \(\frac{1}{2}\).
  • By multiplying both sides of the equation by \(\frac{1}{2}\), or equivalently dividing by \(2\), we can isolate \( x\).
Carrying out this operation:\[ \frac{2x}{2} = \frac{26}{2} \], which simplifies to \( x = 13 \).This mathematical operation is foundational in solving equations as it effectively gives us the value of the variable, confirming that our manipulation has led us to the answer.
Algebraic Manipulation
Algebraic manipulation is the heart of solving equations. It involves systematically using various techniques to simplify and rearrange equations in order to find the solution for the variable. By mastering these techniques, we can tackle both simple and complex equations.To illustrate, let's review the exercise. We started with the equation \(2x - 6 = 20\). Through careful algebraic manipulation:
  • We used the addition inverse to eliminate the constant \(-6\), yielding \(2x = 26\).
  • Next, the multiplicative inverse allowed us to divide by \(2\), resulting in \(x = 13\).
Each step was essential:
  • The addition inverse simplified one side of the equation.
  • The multiplicative inverse isolated "x."
This methodical approach is what makes algebra powerful. Even though the steps might seem simple independently, together they form a full strategy to solve for unknowns effectively and confidently.

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