Problem 61 - Elementary Algebra Within Reach

Question

In Exercises 61-64, solve the equation and check your solution. $$ x-6=5 x $$

Step-by-Step Solution

Answer

The solution to the equation \(x-6=5x\) is \(x = 1.5\). After substituting \(x = 1.5\) into the original equation, it proves that the solution is correct.

Step 1: Combine Like Terms

Move \(x\) from the left side of the equation to the right side to combine the terms containing \(x\). The equation will become \(-4x = 6\).

Step 2: Isolate \(x\)

To isolate \(x\), divide both sides of the equation by -4. The equation will then be \(x = -6 / -4\).

Step 3: Simplify the Right Side

Simplifying the right side of the equation will give the solution for \(x\). Doing the calculation, \(x = 1.5\).

Step 4: Check the Solution

To verify the solution, substitute \(x = 1.5\) into the original equation \(x - 6 = 5x\), and check if the equation holds true. If it does, then the solution is correct.

Key Concepts

Algebra Practice

Algebra practice involves developing your skills in manipulating equations. It's about getting comfortable with rearranging expressions and solving for variables. Practicing regularly helps you develop an intuition for identifying what operations to use and when. To become proficient, you should:

• Work on a variety of problems to expose yourself to different scenarios.
• Understand the fundamental principles of algebra to know why certain steps are taken, like moving terms to different sides of an equation.
• Practice checking your solutions by substituting back into the original equation to ensure they're correct.

Try to gradually increase the complexity of the problems as you improve. Don't rush; mastering each level is important.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions and equations. It involves gathering terms that have the same variable part, which makes the equation easier to handle.When confronting an equation such as \(x - 6 = 5x\), it's essential to recognize that terms involving \(x\) can be moved to one side of the equation. By bringing all the \(x\) terms together, you simplify the equation. In our example, moving \(x\) from the left to the right side results in \(-4x = 6\). Remember, the sign of the term changes when you move it across the equals sign. This technique helps reduce the complexity of the equation, making the next steps clearer.

Isolating Variables

Isolating the variable means getting it by itself on one side of the equation. This is usually done after combining like terms and is essential for finding the solution.In the case of our equation, \(-4x = 6\), we need to isolate \(x\). This is done by performing the inverse operation to what's being applied to \(x\). Here, \(x\) is multiplied by \(-4\), so we divide both sides of the equation by \(-4\) to isolate \(x\). When we do the math, \(x = -6 / -4\), leaving us with \(x = 1.5\). By isolating the variable, we directly find its value, which can then be verified in the original equation.

Simplifying Equations

Simplifying equations is about reducing them to their most basic form without changing their solutions. It's the process of making the equation easier to work with and solve. After isolating the variable, simplify what remains to find a straightforward value for the variable. In our example, after dividing by \(-4\), the equation becomes \(x = 1.5\). Simplifying can also involve factoring out common terms or reducing fractions, as needed. Always check your solution to ensure that the simplification didn't inadvertently change the meaning of the equation. Simplification should always maintain the equality represented by the equation.