Problem 11 - Algebra for College Students

Question

Solve each equation. \(-4=2 x-6\)

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Isolate the Variable Term

Start by isolating the term with the variable on one side of the equation. Add 6 to both sides to move the constant term to the other side: \(-4 + 6 = 2x - 6 + 6\). This simplifies to: \(2 = 2x\).

Step 2: Solve for the Variable

To solve for \(x\), divide both sides of the equation by 2 to get the variable by itself:\(\frac{2}{2} = \frac{2x}{2}\).This simplifies to: \(1 = x\).

Key Concepts

Solving Equations

Solving equations involves finding the value of an unknown variable that makes a given mathematical statement true. In our example, the equation we need to solve is \(-4 = 2x - 6\). The objective is to determine the value of \(x\) such that both sides of the equation are equal. This process usually involves a combination of mathematical operations such as addition, subtraction, multiplication, or division.

• Begin by identifying the terms on both sides of the equation.
• Next, use inverse operations (like adding, subtracting, multiplying, or dividing) to manipulate the terms and find the value of the variable.

As we solve, remember to perform the same operation on both sides to maintain balance, just like keeping a scale level. By simplifying through steps, we can isolate and solve for the variable efficiently.

Isolating Variables

Isolating the variable is a crucial step in solving equations. It means rearranging the equation so that the variable stands alone on one side, making it easier to solve. In the given equation, \(-4 = 2x - 6\), we need to isolate \(x\).
To start, move the constant term (\(-6\)) to the other side of the equation by performing its inverse operation, which is addition. Add 6 to both sides:

• \(-4 + 6\) becomes \(2\).
• The equation updates to \(2 = 2x\).

Now, the variable term \(2x\) is isolated. We've effectively positioned \(x\) for easy solving. This essential step makes the equation workable and simple to resolve.

Equation Simplification

Equation simplification involves reducing the equation to its simplest form so it can be solved more efficiently. Simplification is key in solving equations because it lessens complexity and reveals the easiest path to reach a solution.
In our problem, once \(2x\) is isolated, simplifying it to solve for \(x\) is straightforward. Since we're working with the equation \(2 = 2x\), divide both sides by the coefficient of \(x\), which is 2:

• Divide \(2\) by \(2\) to get \(1\).
• Divide \(2x\) by \(2\) to get \(x\).

Thus, \(x = 1\).
Simplification cuts through unnecessary complexity and directs attention towards achieving the solution, serving as a vital tool in equation-solving techniques.