Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

If \(\frac{x}{5}-2=\frac{x}{3},\) evaluate \(x^{2}-x\)

Step-by-Step Solution

Answer

The value of \(x^{2}-x\) is 210.

Step 1: Solve for \(x\)

Start by solving the equation \(\frac{x}{5}-2=\frac{x}{3}\). Multiply throughout by 15 to remove the denominator, which results in \(3x -30 = 5x\). Then, rearrange this equation to isolate \(x\), giving \(2x = 30\). Solve for \(x\) by dividing both sides by 2, yielding \(x = 15\).

Step 2: Substitute \(x\) into \(x^{2}-x\)

Now that we know \(x =15\), we substitute this value back into the given expression \(x^{2}-x\). So, \(x^{2}-x\) becomes \(15^2-15 = 210\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are symbols put together in a meaningful way using numbers, variables (like \( x \)), and operation symbols such as \( + \) (addition), \( - \) (subtraction), \( * \) (multiplication), and \( / \) (division). These expressions represent quantities without fixed values, often used to describe certain relationships or patterns.

In our exercise, \( x^{2}-x \) is an algebraic expression with \( x \) as the variable. The expression \( x^{2}-x \) can take different values depending on the value of \( x \). The power of two in \( x^{2} \) means \( x \) is squared, while the \( -x \) part means you subtract \( x \) from \( x^{2} \), which represents a simple polynomial where \( x \) is the sole variable.

Understanding how to manipulate these expressions is crucial for solving equations and can be applied to a variety of problems in mathematics.

The Substitution Method

The substitution method is a technique used to solve equations where one variable can be isolated and then used to solve other related equations. This method involves replacing a variable with its corresponding value.

In the provided textbook exercise, the substitution method is applied when we first isolate \( x \) in the equation \( \frac{x}{5}-2=\frac{x}{3} \) and find its value, which turns out to be 15. Then we use this value to substitute \( x \) in the algebraic expression \( x^{2}-x \) to evaluate it.

Steps to Use the Substitution Method

• Isolate the variable in one of the equations (if not already done).
• Substitute the isolated variable's value into the other equation or expression.
• Solve the new equation for the remaining variable, if applicable, or evaluate the expression.

Applying this method systematically can greatly simplify solving numerous types of algebraic problems.

Quadratic Expressions

Quadratic expressions are algebraic expressions of the second degree, meaning the highest power of the variable within the expression is two. The standard form of a quadratic expression is \( ax^{2} + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.

In the context of the given problem, while \( x^{2}-x \) is not a complete quadratic expression since it lacks the constant term \( c \), it still represents a simplified quadratic with \( b = -1 \) and \( c = 0 \). Quadratic expressions are important because they can describe various physical phenomena, such as the trajectory of a projectile or the area of a square.

Characteristics of Quadratic Expressions

• They have a parabolic graph, which is a U-shaped curve.
• The sign of the leading coefficient \( a \) determines whether the parabola opens upwards (if positive) or downwards (if negative).
• The vertex of the parabola represents the maximum or minimum point of the expression, depending on the direction it opens.

Quadratic expressions are solved using methods like factoring, completing the square, or applying the quadratic formula.