Problem 65 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{2.5 x+1}{2}=\frac{4.5}{12} $$

Step-by-Step Solution

Answer

\( x = -\frac{1}{10} \)

Step 1: Cross-Multiply

In a proportion, the cross products are equal. Cross-multiply the terms of the proportion:\[ (2.5x + 1) imes 12 = 4.5 imes 2 \]

Step 2: Simplify the Cross Products

Calculate each side of the equation:- Left side: \[ 12 imes (2.5x + 1) = 30x + 12 \]- Right side: \[ 4.5 imes 2 = 9 \]This gives the equation:\[ 30x + 12 = 9 \]

Step 3: Isolate the Variable Term

Subtract 12 from both sides to isolate the term with the variable:\[ 30x = 9 - 12 \]This simplifies to:\[ 30x = -3 \]

Step 4: Solve for x

Divide both sides by 30 to solve for \(x\):\[ x = \frac{-3}{30} \]Simplify the fraction:\[ x = -\frac{1}{10} \]

Key Concepts

Cross-multiplication

Cross-multiplication is a handy trick to solve proportions, essentially fractions set equal to each other. If you have a fraction on each side of an equation, cross-multiplication helps you eliminate the fractions so you can more easily solve for the variable. To cross-multiply, you take the numerator (top number) of one fraction and multiply it by the denominator (bottom number) of the other fraction. Repeat this step with the other numerator and denominator. Place these products on opposite sides of a new equation.

• Always remember, the equation formed by cross-multiplying will keep the proportions equal.
• Make sure to multiply correctly and carry over any terms included with variables.
• For instance, if you have the proportion \( \frac{2.5x + 1}{2} = \frac{4.5}{12} \), apply cross-multiplication: \( (2.5x + 1) \times 12 = 4.5 \times 2 \).

This gives you a simplified equation free of fractions, making the next steps a lot smoother and easier to navigate.

Isolation of variables

After using cross-multiplication, you are often left with a linear equation on each side of the equal sign. Usually, your goal is to find out what the variable equals, so you'll need to isolate it by itself. To isolate the variable means making the variable term stand alone on one side of the equation. Here's how you can break down the isolation of the variable in our example:

• Subtract any constants from both sides of the equation. For \( 30x + 12 = 9 \), subtract 12 from both sides to leave \( 30x = -3 \).
• To have \( x \) by itself, divide each side by the coefficient of \( x \). In our example, divide by 30 to isolate \( x \).
• This step will give you the value of \( x \) in terms of known numbers.

Once the variable is isolated, you've almost solved the problem! Just one more step is left: simplification if needed.

Simplification of fractions

After isolating the variable, you may often get a fraction as the solution. It’s good to simplify fractions to their most basic form for clarity and precision.Simplifying a fraction means reducing it to the smallest possible numbers in the numerator and denominator without changing its value. Here's how to simplify:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

For instance, in our example, you ended up with \( x = \frac{-3}{30} \). The GCD of 3 and 30 is 3. So, divide both by 3, getting \( x = -\frac{1}{10} \).By simplifying, not only is the solution more readable, but it often gives you better insight, especially in practical applications where simplified results are easier to understand and use.