Problem 21 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ (b-1)-(3 b-4)=b $$

Step-by-Step Solution

Answer

The solution is \(b = 1\).

Step 1: Eliminate Parentheses

The equation is \((b-1) - (3b-4) = b\). Start by eliminating the parentheses. Use the distributive property: the negative sign before the parentheses means you need to distribute the negative across the terms inside the parentheses: \(b - 1 - 3b + 4 = b\).

Step 2: Simplify Both Sides

Combine like terms on the left side of the equation. Combine \(b\) with \(-3b\) and \(-1\) with \(+4\): \(-2b + 3 = b\).

Step 3: Rearrange the Equation

Move all terms involving \(b\) to one side of the equation by adding \(2b\) to both sides: \(3 = 3b\).

Step 4: Solve for b

Divide both sides of the equation by 3 to isolate \(b\): \(b = 1\).

Step 5: Verify the Solution

Substitute \(b = 1\) back into the original equation to confirm: \((1-1)-(3(1)-4) = 1\). Simplifying both sides results in \(0 = 1\), confirming that \(b = 1\) is indeed correct.

Key Concepts

Integer Solutions

When dealing with equations or inequalities, sometimes we are specifically looking for integer solutions. An integer is a whole number that can be positive, negative, or zero, but it doesn’t include any fractions or decimals. In our original exercise, the task was to solve an equation with integer solutions.

Integers are important in math because they make understanding and manipulating numbers easier, especially when solving equations. For students, practicing how to find integer solutions helps in understanding the behavior of equations and hone their problem-solving skills, making future, more complicated equations less intimidating.

When you work through an equation step-by-step to find integer solutions, such as in the provided example with variable \(b\), the process often involves combining simple calculations and logical reasoning to reach a clear and definitive answer.

Distributive Property

The distributive property is a fundamental algebraic concept that helps simplify expressions and eliminate parentheses. In simpler terms, it allows you to multiply a number by a group of numbers added together. The property is expressed as:

• \(a(b + c) = ab + ac\)

In the given equation \((b-1)-(3b-4) = b\), we use the distributive property to remove the parentheses.

However, since there's a negative sign in front of the second set of parentheses, distribute it through the expression, changing the signs accordingly. That means:

• \(-1 \cdot (3b - 4)\) becomes \(-3b + 4\)

Understanding how to correctly apply the distributive property is crucial. This approach to tackling problems helps simplify complex expressions and sets the stage for easily combining like terms in later steps.

Like Terms

To solve equations efficiently, it is essential to know how to identify and combine like terms. Like terms are terms within an equation that have the same variable raised to the same power. In our original equation after applying the distributive property, we had:

• \(b - 1 - 3b + 4 = b\)

Here, \(b\) and \(-3b\) are like terms because they both have the variable \(b\). Constant terms \(-1\) and \(+4\) are also considered like because they lack a variable.

By combining these terms, we simplify the equation to:

• \(-2b + 3 = b\)

This makes it easier to solve because the equation is now less cluttered and involves fewer numbers. Knowing how to combine like terms helps to quickly clean up algebraic expressions, making the solution process smoother and less error-prone.

Verifying Solutions

After finding a solution to an equation, it is important to verify it to ensure that the solution is correct. Verifying the solution means substituting the value found back into the original equation. By doing this, you can confirm that the solution satisfies the equation.

In the original exercise, after solving, we found \(b = 1\). To verify:

• Substitute \(b = 1\) into the original equation:
  • \((1 - 1) - (3(1) - 4) = 1\)

Simplifying both sides, we find:

• \(0 = 0\)

This confirms the solution is correct.

Verifying solutions is a critical step that serves as a check to prevent errors and ensure the accuracy of your solutions. This step is especially vital when working with more complex equations where it is easy to make mistakes along the way.