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Chapter 3: Problem 20

Solve each equation. $$-6 x+1=43$$

Short Answer

Expert verified
The solution is \(x = -7\).

Step by step solution

01

Eliminate the constant term

To begin, we need to isolate the term containing the variable, which is \(-6x\). Start by removing the constant on the left side of the equation by subtracting 1 from both sides:\[-6x + 1 - 1 = 43 - 1\]This simplifies the equation to:\[-6x = 42\]
02

Solve for the variable

Now, we need to solve for \(x\) by getting rid of the coefficient of the variable. Since \(-6\) is multiplied by \(x\), divide both sides of the equation by \(-6\) to isolate \(x\):\[\frac{-6x}{-6} = \frac{42}{-6}\]Thus, we find:\[x = -7\]

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Key Concepts

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

Variable Isolation
To solve equations efficiently, it is crucial to first focus on variable isolation. This means setting up your equation so that the variable, such as \(x\), stands on its own. The goal is to simplify both sides of the equation to have the variable by itself on one side. For instance, in the equation \(-6x + 1 = 43\), the immediate goal is to move any constants away from \(x\) so that the variable can be easily identified and manipulated.
  • Start by identifying non-variable terms on the side of the equation where your variable is.
  • Use addition or subtraction to eliminate these constants.
In our example, we subtract 1 from both sides to achieve \(-6x = 42\), making the variable more isolated. This step is essential as it simplifies the equation significantly, thus making it easier to find the solution.
Solving Equations
At its core, solving equations is the process of finding the value of the variable that makes the equation true. After isolating the variable, the next step involves making the variable pure by executing operations to both sides of the equation.Consider the equation we worked on: after isolating the variable, you have \(-6x = 42\). Here is where you apply division to solve for \(x\) since it is currently being multiplied by \(-6\).
  • Perform the inverse operation, in this case, division.
  • Divide both sides by the coefficient of the variable.
So, dividing both sides by \(-6\) results in \(x = -7\). This step concludes the solving process, giving us the value of \(x\) that satisfies the given equation.
Algebraic Manipulation
Algebraic manipulation involves a series of systematic operations that change the form of an equation while keeping its equality intact. These operations help in both isolating variables and finding their values, as well as simplifying the equations altogether.

Basic Principles

  • Always perform the same operation on both sides of the equation to maintain balance.
  • Use inverse operations to eliminate coefficients and constants.
  • Ensure you simplify your equation step by step for clarity.
In our example, initially, subtraction was used to handle the constant \(+1\) to isolate \(-6x\). Then, division was applied to handle the coefficient \(-6\) to solve for \(x\). Through this process, any equation can be tackled systematically, aiming for clarity and a logical flow that naturally leads to the solution.

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