Problem 38 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 5 b=-2 b+8 b+1 $$

Step-by-Step Solution

Answer

The solution is \(b = -1\).

Step 1: Simplify the Equation

Combine like terms. Start by combining the terms on the right side of the equation. We have \(-2b + 8b\). This simplifies to \(6b\). So the equation becomes: \[ 5b = 6b + 1 \]

Step 2: Isolate the Variable

To solve for \(b\), we must get \(b\) on one side of the equation. We can do this by subtracting \(6b\) from both sides of the equation: \[ 5b - 6b = 6b + 1 - 6b \]This simplifies to:\[ -b = 1 \]

Step 3: Solve for the Variable

The equation \(-b = 1\) implies that \(b\) is the opposite of 1. To find \(b\), multiply both sides by \(-1\):\[ b = -1 \]

Step 4: Check the Solution

Substitute \(b = -1\) back into the original equation to confirm it's correct:\[ 5(-1) = -2(-1) + 8(-1) + 1 \]This calculates to:\[-5 = 2 - 8 + 1 \]Which simplifies to:\[-5 = -5 \]The solution is correct.

Key Concepts

Combining Like Terms

In algebra, combining like terms is an essential step for simplifying equations. Like terms are terms that have the same variable raised to the same power. For example, the terms \(-2b\) and \(8b\) are like terms because they both have the variable \(b\) raised to the power of 1. To combine them, sum their coefficients while maintaining the variable part.

Here's how it works:

• Start with the terms: \(-2b + 8b\)
• Combine them by adding the coefficients: \( -2 + 8 = 6\)
• The result is \(6b\), which replaces \(-2b + 8b\)

This step simplifies the equation from \(5b = -2b + 8b + 1\) to \(5b = 6b + 1\). Combining like terms makes it easier to handle and solve equations.

Isolating the Variable

To find the value of a variable in an equation, we need to isolate it on one side of the equation. This is a fundamental algebraic technique that helps us "see" the value of the variable more clearly.

Here’s the process:

• Start with the simplified equation: \(5b = 6b + 1\)
• Subtract \(6b\) from both sides to get rid of \(b\) on the right side: \(5b - 6b = 1\)
• This leaves us with \(-b = 1\)

To solve for \(b\), multiply by \(-1\) to convert \(-b\) to \(b\).

• Multiply both sides: \(-b \times -1 = 1 \times -1\)
• The solution becomes \(b = -1\)

By isolating \(b\), we have determined its value as \(-1\). This process is also referred to as "solving for the variable."

Checking Solutions in Equations

After finding a solution, it's crucial to verify that it's correct by substituting the value back into the original equation. This step ensures that no mistakes were made during the process.

To check our solution \(b = -1\):

• Substitute \(-1\) for \(b\) in the original equation: \(5(-1) = -2(-1) + 8(-1) + 1\)
• Simplify the left side: \(-5\)
• Simplify the right side: \(2 - 8 + 1 = -5\)

Both sides of the equation equal \(-5\), confirming that \(b = -1\) is indeed the correct solution. Double-checking solutions by substitution is an effective way to validate your work and ensure accuracy. Always take this step to avoid potential errors.