Problem 65 - Elementary Algebra

Question

Solve. $$ -8=4-2 x $$

Step-by-Step Solution

Answer

The solution is \(x = 6\).

Step 1: Move the constant term

To isolate the term containing the variable, we need to move the constant term from the right-hand side to the left-hand side of the equation. Start by subtracting 4 from both sides:\(-8 - 4 = 4 - 2x - 4\)This simplifies to:\(-12 = -2x\)

Step 2: Eliminate the coefficient of x

Next, we need to isolate \(x\) by getting rid of the coefficient \(-2\). We will do this by dividing both sides of the equation by \(-2\):\[\frac{-12}{-2} = \frac{-2x}{-2}\]This simplifies to:\(6 = x\)

Step 3: Simplify and rearrange

Finally, rewrite the equation with \(x\) on the left side to state the solution clearly:\(x = 6\)

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions that represent a line when graphed. They follow the general format of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
Solving these equations helps us find the value of \(x\) that makes the equation true. In the exercise provided, we need to determine what value of \(x\) makes the equation \(-8 = 4 - 2x\) valid.
To solve a linear equation like this, we systematically use operations like addition, subtraction, multiplication, and division. These operations simplify the equation, helping to isolate \(x\) and ultimately solve for it.

Step by Step Solutions

Taking a step-by-step approach to solve equations is crucial for clarity and accuracy. It ensures that we don't miss any important operations.
Breaking down the problem into smaller steps:

• Helps identify and address each part of the equation effectively.
• Makes it easier to catch and correct mistakes, leading to a correct solution more often.
• Provides a logical flow, offering insights into why each operation is performed.

In the provided example, we solve the problem by first removing constants and then dealing with coefficients to simplify the equation progressively.
This clear structure ensures that each step logically leads to the next, making the problem-solving process approachable.

Isolation of Variable

Isolating the variable is a central aspect of solving equations. It involves using algebraic operations to 'isolate' the variable \(x\) on one side of the equation, leaving it alone.
This method requires:

• Transposing terms across the equation by performing equal operations on both sides to maintain balance.
• Removing the coefficient through division or multiplication so that the variable is by itself.

For instance, in our original problem, \(-8 = 4 - 2x\), we moved the constant 4 to the left, followed by dividing by \(-2\) to get \(x\) alone.
This process helps clearly display the solution, making it easier to understand and verify.

Simplifying Equations

Simplification is the process of making equations easier to work with by reducing complexity. It involves combining like terms, reducing fractions, or eliminating unnecessary components.
In our example:

• We first combined like terms by subtracting 4 from both sides, transforming \(-8 - 4 = 4 - 2x - 4\) into \(-12 = -2x\).
• Then, the equation was simplified further by eliminating the coefficient of the variable through division, rendering it to its final form \(x = 6\).

Each simplification step is crucial as it incrementally solves the equation and often reveals insightful patterns or relationships.