Problem 68 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10 .\) See Example 6. $$ \frac{|5 y-x|}{6 t} $$

Step-by-Step Solution

Answer

The value of the expression is \(\frac{5}{12}\).

Step 1: Identify the expression

We need to evaluate the expression \(\frac{|5y-x|}{6t}\). Let's substitute the values \(x = -5\), \(y = 4\), and \(t = 10\) into the expression.

Step 2: Substitute the values

Replace \(x\) with \(-5\), \(y\) with \(4\), and \(t\) with \(10\) in the expression: \[\frac{|5(4) - (-5)|}{6(10)}\]

Step 3: Simplify the numerator

Calculate the expression inside the absolute value: \[5 \times 4 = 20\]Then add \(5\) (since subtracting \(-5\) is the same as adding \(5\)): \[20 + 5 = 25\]Thus, the expression becomes:\[\frac{|25|}{6 \times 10}\]

Step 4: Simplify the denominator

Calculate the denominator: \[6 \times 10 = 60\]Our expression simplifies to:\[\frac{25}{60}\]

Step 5: Simplify the fraction

Simplify the fraction \(\frac{25}{60}\) by finding the greatest common divisor (GCD) of \(25\) and \(60\), which is \(5\):\[\frac{25 \div 5}{60 \div 5} = \frac{5}{12}\]

Step 6: Write the final answer

The value of the expression \(\frac{|5y-x|}{6t}\) when \(x=-5\), \(y=4\), and \(t=10\) is \(\frac{5}{12}\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. It reveals how big a number is, disregarding any negative sign. When evaluating absolute values, such as in the expression \(|5y - x|\), it's critical to deal with what's inside before applying the rule of absolute value.

• Calculate the inner expression: First, determine the result of \((5 \times 4) - (-5)\) which equals \((20 + 5)\).
• Apply the absolute value: The result of \(25\) is already positive, so \(|25|\) equals \25\.

Understanding absolute value helps simplify expressions, ensuring the number inside is handled correctly.

Substitution in Expressions

Substitution is replacing variables in an expression with their given values. For instance, in the expression \(|5y-x|/(6t)\), you'll substitute the variables with their values. Here's how you do it step by step:

• Identify the given values: \(x = -5\), \(y = 4\), and \(t = 10\).
• Substitution process: Replace each variable in the expression with these values, resulting in \(\frac{|5(4) + 5|}{6(10)}\).
• Calculate: Carry out all the multiplication and addition involved, leading to \(\frac{|25|}{60}\).

This systematic approach to substitution helps in breaking down and simplifying more complex mathematical expressions and solving problems accurately.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. To simplify, find the greatest common divisor (GCD) of the numerator and denominator. Here's how to apply it:

• Identify the fraction: \(\frac{25}{60}\).
• Calculate the GCD: The greatest number that divides both \(25\) and \(60\) is \(5\).
• Divide both terms: Doing so gives \(\frac{25 \div 5}{60 \div 5} = \frac{5}{12}\).
• Simplified Result: Your fraction \(\frac{5}{12}\) cannot be simplified further since \(5\) and \(12\) have no common factors besides \(1\).

Simplifying fractions ensures you leave no redundancies in your final answer, making the result easy to understand and use in further calculations.