Problem 62 - Algebra A Combined Function

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10\). \(\frac{15-x}{y+2}\)

Step-by-Step Solution

Answer

The value of the expression is \( \frac{10}{3} \).

Step 1: Substitute the Values

First, we substitute the given values for the variables in the expression. The expression is \( \frac{15-x}{y+2} \). Given \( x = -5 \), \( y = 4 \), we have: \[ \frac{15 - (-5)}{4 + 2} \]

Step 2: Simplify the Numerator

Now, simplify the numerator of the expression \( 15 - (-5) \). Notice that subtracting a negative is equivalent to adding the positive, so: \[ 15 - (-5) = 15 + 5 = 20 \]

Step 3: Simplify the Denominator

Simplify the denominator by adding the values together: \[4 + 2 = 6\]

Step 4: Divide the Numerator by the Denominator

Finally, divide the simplified numerator by the simplified denominator: \[\frac{20}{6}\] To further simplify, divide both terms by their greatest common divisor, which is 2:\[\frac{20 \, \div \, 2}{6 \, \div \, 2} = \frac{10}{3}\]

Key Concepts

Substitution

Substitution in algebra is like filling in the blanks. You are given values for variables and you replace the variables in the given expression with these values. It can simplify understanding what the expression equates to.
When using substitution, you directly replace the letters in an expression with the specific numbers they represent. This helps in transforming the expression into a more solvable arithmetic calculation.

• Ensure all given values are correctly substituted without forgetting any part of the expression.
• Be careful with negative signs when substituting; remember that subtracting a negative number turns into addition.

For example, in the expression \(\frac{15-x}{y+2}\), substitute \(x = -5\) and \(y = 4\). This becomes \(\frac{15-(-5)}{4+2}\).

Simplifying Expressions

Simplifying means making an expression easier to work with or understand by reducing its complexity. This usually involves performing arithmetic operations like addition or subtraction directly.
In algebra, simplifying can involve combining like terms or performing operations. With our example, simplify the numerator and the denominator separately.

• For a numerator like \(15 - (-5)\), note subtraction of a negative number is the same as adding the number - hence \(15 + 5 = 20\).
• The denominator \(4 + 2 = 6\) can be simplified by performing the addition.

After substitution and simplification, the expression becomes \(\frac{20}{6}\). It's much easier to handle from here.

Numerator and Denominator

The structures of fractions, numerators and denominators, are essential to understanding and simplifying algebraic expressions. The fraction line signifies division, and the numerator is the top number, while the denominator is the bottom one.
The numerator represents how many parts we have, whereas the denominator shows into how many parts the whole is divided. With \(\frac{20}{6}\), 20 is the numerator, and 6 is the denominator.

• Always simplify the numerator and the denominator when possible, before concluding with division. It makes the division step straightforward, ensuring calculations are simpler.
• Keep in mind the mathematical operations applied on both parts separately.

Handling each part of the fraction effectively leads to easier problem-solving.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can exactly divide two numbers without leaving a remainder. It is a helpful tool when you want to simplify fractions to their lowest terms.
Finding the GCD involves identifying numbers that divide evenly into both the numerator and the denominator.

• For the example \(\frac{20}{6}\), both 20 and 6 can be divided by 2, the GCD of 20 and 6, giving \(\frac{10}{3}\).
• Divide both the numerator and denominator by their GCD to simplify the fraction.

When simplified correctly using the GCD, expressions can be much clearer and easier to work with.