Problem 22 - Elementary Algebra

Question

Solve. $$ 5(4 y+3)=0 $$

Step-by-Step Solution

Answer

\( y = \frac{-3}{4} \)

Step 1 - Simplify the Equation

Start by distributing the 5 to both terms inside the parentheses. This gives: \[ 5 \times 4y + 5 \times 3 = 0 \]. Simplifying this, we get: \[ 20y + 15 = 0 \]

Step 2 - Isolate the Variable Term

To isolate the term with the variable, subtract 15 from both sides of the equation: \[ 20y + 15 - 15 = 0 - 15 \]. This simplifies to: \[ 20y = -15 \]

Step 3 - Solve for y

Finally, divide both sides of the equation by 20 to solve for \( y \). This gives us: \[ y = \frac{-15}{20} \]. Simplify the fraction: \[ y = \frac{-3}{4} \]

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra. It helps you expand expressions so you can simplify and solve equations. The distributive property states that for any numbers, \(a\), \(b\), and \(c\): \(a(b + c) = ab + ac\). In our example problem, the expression \(5(4y + 3)\) uses the distributive property.
To apply it, we distribute the 5 to both terms inside the parentheses:

• First term: \(5 \times 4y = 20y\)
• Second term: \(5 \times 3 = 15\)

This gives us the equation \(20y + 15 = 0\). Now, the equation is easier to work with because it's simpler and all the terms are expanded.

isolate variable

Now that we have \(20y + 15 = 0\), it's time to isolate the variable \(y\). To isolate means to get the variable alone on one side of the equation.
This process often involves a couple of steps. In our case, the first step is to eliminate the constant term on the same side as \(y\).
We subtract 15 from both sides of the equation:

• \(20y + 15 - 15 = 0 - 15\)
• This gives us \(20y = -15\)

We are getting closer to isolating \(y\). The next step is to get \(y\) by itself by dividing both sides by the coefficient of \(y\), which is 20:
This gives us: \(y = \frac{-15}{20}\)

simplifying fractions

The last part of solving the equation involves simplifying fractions. The fraction \(\frac{-15}{20}\) can be simplified. To simplify fractions, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 15 and 20 is 5. We divide both the numerator and the denominator by their GCD:

• \(\frac{-15 \text{ \textbackslash}div 5}{20 \text{ \textbackslash}div 5} = \frac{-3}{4}\)

This simplified fraction means that \(y = \frac{-3}{4}\).
Simplifying fractions makes your answer more manageable and easier to understand.
Remember: always look for common factors to simplify fractions as much as possible.