Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$x+3 y=0 \quad(0,0),\left(1, \frac{1}{3}\right),\left(2,-\frac{2}{3}\right)$$

Step-by-Step Solution

Answer

Only the ordered pairs (0,0) and (2, -\(\frac{2}{3}\)) are solutions to the equation.

Step 1: Substitute the coordinates from the ordered pair (0,0) into the equation

For the pair (0,0), replace x with 0 and y with 0 in equation \(x+3y=0\). This gets us \(0+3(0)=0\). Simplifying, we get on the left-hand side 0, which is equal to the right-hand side. So, pair (0,0) is a solution.

Step 2: Substitute the coordinates from the ordered pair (1,\(\frac{1}{3}\)) into the equation

For the pair (1,\(\frac{1}{3}\)), replace x with 1 and y with \(\frac{1}{3}\) in equation \(x+3y=0\). This gets us \(1+3\left(\frac{1}{3}\right)=0\). Simplifying, we get on the left-hand side 2, which is different from 0. Therefore, pair (1,\(\frac{1}{3}\)) is not a solution.

Step 3: Substitute the coordinates from the ordered pair (2, -\(\frac{2}{3}\)) into the equation

For the pair (2, -\(\frac{2}{3}\)), replace x with 2 and y with -\(\frac{2}{3}\) in equation \(x+3y=0\). This gets us \(2+3(-\frac{2}{3})=0\). Simplifying, we get on the left-hand side 0, which is equal to the right-hand side. So, pair (2,-\(\frac{2}{3}\)) is a solution.

Key Concepts

Ordered Pairs

An ordered pair typically refers to two numbers arranged in a specific, designated order. In the context of coordinate geometry, these numbers represent coordinates that pinpoint the location of a point on a two-dimensional plane, with the convention of using \( (x, y) \) to denote the x-coordinate and y-coordinate, respectively.

When solving algebraic equations like \( x+3y=0 \), you can test whether an ordered pair is a solution by substituting the values of the pair into the equation. If the equation holds true, meaning the left- and right-hand sides are equal after the substitution, then the ordered pair is a valid solution to the equation.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. A linear equation in two variables, such as \( x \) and \( y \), has the general form \( Ax + By = C \), where \( A \) and \( B \) are coefficients and \( C \) is the constant term.

The equation \( x+3y=0 \) in the exercise is a simple example of a linear equation. Linear equations are notable for their predictability and ease of solution using methods such as graphing, substitution, or elimination. They represent a foundational concept in the study of algebra and are frequently used to determine the relationships between two variables.

Substitution Method

The substitution method is a strategy used for solving systems of equations as well as verifying solutions to an equation. This involves replacing variables with their corresponding values and thereby simplifying the equation.

In our example, to determine whether an ordered pair is a solution to the given linear equation, we substitute the x and y values from the pair into the equation and check the validity of the resulting expression. For instance, substituting the pair \( (0,0) \) into the equation \( x+3y=0 \) simplifies to \( 0+3(0)=0 \) confirming that \( (0,0) \) is indeed a solution.

Algebraic Solutions

Algebraic solutions are the results we obtain after correctly applying algebraic methods, like substitution, to solve equations. These solutions can be numbers, ordered pairs, or more complex structures, and they must satisfy the original equation.

For example, after testing the ordered pairs \( (0,0) \) and \( (2,-\frac{2}{3}) \) in the exercise, we found that they satisfied the equation \( x+3y=0 \) and were thus algebraic solutions. On the other hand, the ordered pair \( (1, \frac{1}{3}) \) did not satisfy the equation, signifying that not all tested pairs will necessarily be solutions. Algebraic solutions provide much-needed confirmation for the values that equations can work with.