Problem 11 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$6 x-8=-x-8$$

Step-by-Step Solution

Answer

The solution to the equation is \\(x = 0\\).

Step 1: Simplify Both Sides

First, we simplify the equation by removing terms that are the same on both sides. Here, we notice that \(-8 = -8\), so we can ignore these on both sides of the equation, giving us the new equation: \6x = -x\.

Step 2: Move Variables to One Side

Next, we want to isolate terms with the variable on one side of the equation. We do this by adding \(x\) to both sides of the equation: \6x + x = 0\.

Step 3: Combine Like Terms

Combine the terms on the left side: \(6x + x = 7x\). So the equation becomes \7x = 0\.

Step 4: Solve for x

Finally, solve the equation \(7x = 0\) by dividing both sides by 7: \x = \frac{0}{7} = 0\.

Key Concepts

Solving Equations

To solve an equation means to find the value of the variable that makes the equation true. It's like unraveling a puzzle where you look for the particular number that, when substituted for the variable, balances both sides of the equation. Here are a few basic steps to keep in mind when solving linear equations:

• Ensure you have a clear equation with terms on both sides.
• First, simplify each side by combining like terms.
• Use inverse operations—like addition to cancel out subtraction—to get the variable by itself.
• Frequently check your work by substituting the solution back into the original equation.

Think of solving an equation as a game of balance where every move you make on one side must also be made on the other. This symmetrical approach keeps the equation "balanced" throughout the solution process.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. This process can help you see the relationships between different parts of an equation clearly. In the context of our original problem, simplifying means getting rid of unnecessary terms and combining like terms:

• Identify terms that can cancel each other out (like the \(-8\) on both sides of the equation in this exercise).
• Combine any like terms, such as variables or constants, to streamline the equation.
• Make sure that what's left is as straightforward as possible so that you can easily solve for the unknown variable.

Simplifying is crucial because it reduces the complexity of the equation, making subsequent steps, like isolating the variable, easier to manage. It’s like clearing out the clutter so you can focus on the core problem.

Isolating Variables

When you isolate the variable in an equation, you rearrange the equation so that the variable stands alone on one side with a coefficient of 1. This is usually done towards the end of solving linear equations, especially when you are left with an equation like \(7x = 0\).

• First, make sure all variable terms are on one side of the equation and all constants are on the other.
• Combine like terms if necessary after moving them across the equal sign.
• If the variable has a coefficient other than 1, divide every term by that coefficient to solve for the variable.

In our example, dividing both sides of \(7x = 0\) by 7 gives \(x = 0\). This means the solution is simple once the variable is isolated, highlighting the value of clear, methodical steps throughout the process. Isolating the variable transforms a complex equation into a simple statement of equality, making the solution evident.