Problem 53 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{b+4}{5}=\frac{3 b-6}{3} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( b = \frac{7}{2} \).

Step 1: Cross-Multiply

In a proportion, you can solve for the variable by cross-multiplying. This means you multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.So, we have:\( (b + 4) imes 3 = 5 imes (3b - 6) \)

Step 2: Distribute and Simplify

Distribute the numbers across the terms in each parenthesis:\( 3(b + 4) = 5(3b - 6) \)\( 3b + 12 = 15b - 30 \)

Step 3: Rearrange the Equation

Get all terms with the variable on one side and constants on the other side of the equation:Subtract \(3b\) from both sides:\( 12 = 12b - 30 \)

Step 4: Isolate the Variable

Add 30 to both sides to isolate the terms with the variable:\( 42 = 12b \)Now, divide both sides by 12 to solve for \(b\):\( b = \frac{42}{12} \)

Step 5: Simplify the Solution

Simplify \( \frac{42}{12} \) by dividing the numerator and the denominator by their greatest common divisor, which is 6:\( b = \frac{7}{2} \)

Key Concepts

Cross-Multiplication

Cross-multiplication is a powerful tool for solving proportions, which are simply equations where two fractions are set equal to one another. By using cross-multiplication, we eliminate the fractions, making it easier to work with the equation. It involves multiplying the numerator of one fraction by the denominator of the other, setting these products equal. For the equation \( \frac{b+4}{5} = \frac{3b-6}{3} \), cross-multiplication gives: \[ (b + 4) \times 3 = 5 \times (3b - 6) \] This step is crucial as it helps us change the fractional equation into a linear form. This method only works because we assume that the value of the two fractions is equal, allowing the products of the means and extremes to be the same. As a student, always remember this step when dealing with proportions as it's the first gateway to simplifying and solving the equation.

Distributive Property

The distributive property is a useful property of numbers that helps to simplify expressions. It tells us how to multiply a number by a group of numbers added together. For our problem, we apply it after cross-multiplication to help get rid of parentheses. In the equation \( 3(b + 4) = 5(3b - 6) \), we distribute the number outside the parenthesis to each term within the parenthesis:

• \( 3 \times b + 3 \times 4 \Rightarrow 3b + 12 \)
• \( 5 \times 3b - 5 \times 6 \Rightarrow 15b - 30 \)

By distributing the factors, we transform the equation to simpler terms like \( 3b + 12 = 15b - 30 \), making it much easier to handle. Mastery of the distributive property can streamline complex multiplications and is incredibly handy when dealing with algebraic expressions with parentheses.

Algebraic Equations

Once you eliminate fractions and distribute terms, you will often be left with an algebraic equation. In the example, the equation has transformed to \( 3b + 12 = 15b - 30 \). Solving algebraic equations involves isolating the variable on one side:

• First, simplify both sides if needed. In our case, this has already been done through distribution.
• Rearrange the equation to gather all terms involving the variable on one side, and constants on the other. Subtract \( 3b \) from both sides to obtain \( 12 = 12b - 30 \).
• Add or subtract constants to isolate the variable term \( (42 = 12b) \).
• Finally, divide by the coefficient of the variable \( (b = \frac{42}{12}) \).

Solving these equations step by step helps you identify the correct value of the variable, making these equations vital in algebra. Simplifying the final answer to \( b = \frac{7}{2} \) ensures that we have the solution in its simplest form. Understanding each stage helps in tackling more complex algebraic problems.