Problem 101 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(\frac{5 y^{2}+4}{x}\)

Step-by-Step Solution

Answer

The value of the expression is 14.

Step 1: Identify given variables

The exercise gives the values for variables: \(x = 6\), \(y = -4\), and \(a = 3\).

Step 2: Substitute the values into the expression

Take the expression \(\frac{5 y^{2} + 4}{x}\). Substitute \(y = -4\) and \(x = 6\) into the expression.

Step 3: Calculate the numerator

First, square the \(y\) value and multiply by 5: \(-4^2 = 16\). So, \(5 \cdot 16 = 80\).Add the remaining constant term: \(80 + 4 = 84\). The numerator is 84.

Step 4: Calculate the entire expression

Now, divide the numerator by \(x\): \(\frac{84}{6} = 14\).

Key Concepts

substitution method

The substitution method is a powerful tool in algebra.
It's used to replace variables in an expression with their given values.
Here's how it works.
First, identify the values you need to substitute.
In this problem, we have:

• x = 6
• y = -4
• a = 3

Next, take the expression you're given.
For this exercise, it is \(\frac{5 y^{2} + 4}{x}\).
Now substitute y = -4 and x = 6 in place of the variables in the expression.
This step ensures the expression is transformed into one that only has numbers, making it easier to calculate.

numerator calculation

Numerator calculation is key when dealing with fractions.
The numerator is the top part of a fraction.
Let's focus on calculating the numerator here.
The expression provided is \(\frac{5 y^{2} + 4}{x}\).
After substituting y = -4, the numerator becomes \(5 (-4)^{2} + 4\).
Follow these steps for the calculation:

• First, square the substituted value of y. Here, \((-4)^{2} = 16\).
• Then, multiply this squared value by 5. So, \(5 \times 16 = 80\).
• Finally, add 4 to the result. Therefore, \(80 + 4 = 84\).

So, the numerator is 84.
This value is now ready to be used in the next step of solving.

simplifying expressions

Simplifying expressions is an essential skill in algebra.
It makes complex expressions easier to work with.
Once you’ve calculated the numerator, the next step is to simplify the entire expression.
The full expression we're working with is \(\frac{5 y^{2} + 4}{x}\).
We already calculated the numerator as 84.
Now, substitute the value of x = 6 in the denominator.
So your expression becomes \( \frac{84}{6} \).
To simplify, divide the numerator (84) by the denominator (6).

• The result of \( \frac{84}{6} \) is 14.

Therefore, the simplified expression evaluates to 14.
Breaking down complex expressions step-by-step makes them far more manageable and easier to understand.