Problem 44 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{3 x}{16}=\frac{x+2}{5} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( x = -32 \).

Step 1: Set up the cross multiplication

To solve the proportion \( \frac{3x}{16} = \frac{x+2}{5} \), start by cross-multiplying the terms. This means multiplying the outer terms and setting them equal to the product of the inner terms. We have:\[ 3x \cdot 5 = 16 \cdot (x + 2) \]

Step 2: Simplify both sides

First, simplify the left side:\[ 15x \]Now, expand the right side distribution:\[ 16 \times x + 16 \times 2 = 16x + 32 \]

Step 3: Rearrange the equation

Set the equation from Step 2:\[ 15x = 16x + 32 \]Subtract \( 16x \) from both sides to isolate the terms with \( x \):\[ 15x - 16x = 32 \]

Step 4: Simplify

Simplify the left-hand side by combining like terms:\[ -x = 32 \]

Step 5: Solve for x

Divide each side by \(-1\) to find the value of \( x \):\[ x = -32 \]

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method used to solve proportions. Proportions are equations that state two ratios or fractions are equal. For example, if you have an equation like \( \frac{3x}{16} = \frac{x + 2}{5} \), you can use cross multiplication to eliminate the fractions and simplify the equation.
The cross multiplication method involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. This gives you a new equation without fractions. In our example, you multiply \( 3x \) by \( 5 \) and then \( 16 \) by \( x+2 \), resulting in the equation:

• \( 3x \cdot 5 = 16 \cdot (x + 2) \)

This equation can be easier to work with because it doesn't include fractions and allows direct manipulation of algebraic terms. Cross multiplication transforms a proportion into a solvable linear equation.

Algebraic Equations

Algebraic equations are mathematical statements that use variables, numbers, and operations to denote a relationship between elements. The equation derived from cross multiplication, such as \( 15x = 16x + 32 \) from our problem, is a linear equation. Solving these equations involves performing operations to simplify the expression and isolate the variables.
In our context, after cross multiplying, we expand and simplify each side of the equation:

• Left side: \( 3x \cdot 5 = 15x \)
• Right side: \( 16(x + 2) = 16x + 32 \)

Algebraic equations often require techniques such as distributing multiplication over addition or combining like terms. Each term in the equation should be managed carefully to ensure that the final equation remains balanced.

Isolating Variables

One of the final steps in solving algebraic equations is isolating the variable, which means getting the variable alone on one side of the equation while all constants (numbers) are on the other side.
In our example, we started with the equation \( 15x = 16x + 32 \):

• Subtract \( 16x \) from both sides to get \( 15x - 16x = 32 \)
• This simplifies to \( -x = 32 \)

Now, to solve for \( x \), we need to divide both sides by \(-1\):

• \( x = -32 \)

Isolating variables is crucial because it gives a clear value for the variable, helping you find the solution to the problem. Every step aims to simplify the equation until the variable is isolated, thus solving the equation.