Problem 1 - Prealgebra

Question

Solve equation. Check your solution. \(4 x-8=5 x\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = -8\).

Step 1: Start with the original equation

We are given the equation \(4x - 8 = 5x\). To solve for \(x\), we need to isolate \(x\) on one side of the equation.

Step 2: Rearrange the equation

Subtract \(4x\) from both sides of the equation to get the \(x\)-terms on one side. This gives us: \(-8 = x\).

Step 3: Simplify

Now that the equation is simplified, we have \(-8 = x\), which means \(x = -8\).

Step 4: Check the solution

To verify the solution, substitute \(x = -8\) back into the original equation: \(4(-8) - 8 = 5(-8)\). This simplifies to \(-32 - 8 = -40\) and \(-40 = -40\), confirming our solution is correct.

Key Concepts

Isolation of Variables

When solving equations, a critical skill is isolating the variable you are solving for. This means getting the variable alone on one side of the equation. This step sets the stage for finding the value of the unknown variable.

In the exercise you encountered, the equation was given as:

• 4x - 8 = 5x

To isolate the variable, you need to have all terms involving x on one side of the equation and all constant terms on the other. You start by subtracting 4x from both sides, which allows you to concentrate the x-terms. This step transforms the equation into:

• -8 = x

Now, x is isolated, and you can easily see that x equals -8.

This strategy works because it uses basic properties of equality: if you do the same operation on both sides of the equation, the balance is maintained. By rearranging terms and performing arithmetic operations, you can achieve an isolated variable, paving the way for finding its value.

Equation Verification

After isolating variables and solving an equation, it is vital to check your solution to ensure its accuracy. This process is known as equation verification. Verification confirms that your solution satisfies the original equation.

In the previous example, you found that x equals -8. To verify, substitute this value back into the original equation:

• 4x - 8 = 5x

By replacing x with -8, evaluate both sides:

• 4(-8) - 8
• 5(-8)

Both expressions simplify to -40, confirming they are equal, and verifying that the solution x = -8 is indeed correct.

Verification acts as a safety check. It helps you confirm that no arithmetic mistakes were made during isolation or simplification. This is crucial for ensuring your solution's reliability and accuracy.

Substitution in Equations

Substitution is a powerful technique used frequently in solving equations, especially during verification. It involves replacing a variable with a known value to simplify and check equations.

In the problem at hand, you used substitution to verify your solution. Once you determined that x equals -8, you substituted this value back into the original equation to check if both sides equaled each other:

• 4(-8) - 8 = 5(-8)

By substituting x with -8, and simplifying both sides, you arrived at the equation -40 = -40.

Substitution works not only for verification but also as a part of problem-solving when dealing with expressions containing more than one variable. By substituting known values, you can break down complex equations into simpler forms and solve for unknowns more easily.