Problem 45 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(35-46,\) determine which, if any, of the ordered pairs listed satisfy the given equation. $$y=\frac{1}{x+2} ; \quad(0,2),\left(1, \frac{1}{3}\right),(-1,1)$$

Step-by-Step Solution

Answer

The ordered pairs \((1, \frac{1}{3})\) and \((-1, 1)\) satisfy the equation.

Step 1: Understand the given equation

The given equation is a rational function: \[ y = \frac{1}{x+2} \]This equation describes a relationship between variable x and variable y.

Step 2: Check the first ordered pair \(\boldsymbol{(0,2)}\)

Substitute \(x = 0\) into the equation: \[ y = \frac{1}{0+2} = \frac{1}{2} \] The resulting \(y\) value is \(\frac{1}{2}\), not \(2\). Therefore, the ordered pair \((0,2)\) does not satisfy the equation.

Step 3: Check the second ordered pair \(\boldsymbol{(1, \frac{1}{3})}\)

Substitute \(x = 1\) into the equation: \[ y = \frac{1}{1+2} = \frac{1}{3} \]This matches the \(y\) value of the ordered pair \((1, \frac{1}{3})\). So, the ordered pair \((1, \frac{1}{3})\) satisfies the equation.

Step 4: Check the third ordered pair \(\boldsymbol{(-1, 1)}\)

Substitute \(x = -1\) into the equation: \[ y = \frac{1}{-1+2} = \frac{1}{1} = 1 \]This matches the \(y\) value of the ordered pair \((-1, 1)\). So, the ordered pair \((-1, 1)\) satisfies the equation.

Key Concepts

Rational Functions

A rational function is a type of function represented by the ratio of two polynomials. The general form of rational functions is: \[ f(x) = \frac{P(x)}{Q(x)} \] where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) cannot be zero because division by zero is undefined. In our exercise, the rational function given is: \[ y = \frac{1}{x + 2} \] Here, the numerator is 1 (a constant), and the denominator is a linear polynomial \(x + 2\).

Rational functions can have vertical asymptotes, which occur where the denominator equals zero. For our function \( y = \frac{1}{x+2} \), the vertical asymptote is at \(x = -2\). Understanding these properties helps us analyze and predict the behavior of the function better.

Ordered Pairs

An ordered pair is a set of numbers used to represent a point in a coordinate system. It consists of two components:\( (x, y) \), where \(x\) represents the horizontal position, and \(y\) represents the vertical position. For instance, in the ordered pair \( (1, \frac{1}{3}) \), 1 is the x-coordinate and \(\frac{1}{3}\) is the y-coordinate.

When dealing with equations, ordered pairs are used to verify if the given pair satisfies the equation. This means substituting the x and y values into the equation and checking if the equation holds true. If it does, the pair is a solution to the equation.

• The pair \( (1, \frac{1}{3}) \) satisfies \( y = \frac{1}{x+2} \) because substituting \(x = 1\) gives \( y = \frac{1}{3} \).
• The pair \( (0, 2) \) does not satisfy \( y = \frac{1}{x+2} \) because substituting \(x = 0\) gives \( y = \frac{1}{2} \).
• The pair \( (-1, 1) \) satisfies the equation because substituting \(x = -1\) gives \( y = 1 \).

Understanding ordered pairs and how to check them against an equation is crucial for solving algebraic problems accurately.

Substitution Method

The substitution method is a technique used to solve equations by replacing variables with their corresponding values. Let's see how we applied this method in the exercise:

• First, we took the ordered pair \((0, 2)\). We substituted \( x = 0 \) into the equation \( y = \frac{1}{x+2} \) to see if the resulting \( y \) value matches 2. It turned out to be \( \frac{1}{2} \), so this pair does not satisfy the equation.
• Next, we took the pair \((1, \frac{1}{3}) \). We substituted \( x = 1 \) and found \( y = \frac{1}{3} \). It matches, so this pair satisfies the equation.
• Lastly, for the pair \((-1, 1)\), substituting \( x = -1\) gave \( y = 1 \), showing this pair satisfies the equation.

Using the substitution method allows us to systematically check if an ordered pair satisfies an equation by plugging in the values and simplifying the equation. It's a fundamental technique in algebra that helps us demonstrate and verify solutions effectively.