Problem 48 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$y+2=0 \quad(0,2),(2,0),(0,-2)$$

Step-by-Step Solution

Answer

The only solution among given pairs for the equation \(y+2=0\) is (0,-2).

Step 1: Identify Ordered Pairs

Identify the y-value in each of the given ordered pairs. The pairs are (x, y) format. The pairs are (0,2), (2,0), and (0,-2). So the y-values are 2, 0, and -2, respectively.

Step 2: Substitute the y-value into the equation

The equation is \(y+2=0\). Substitute the y-values from each pair into the equation and check if equality is maintained. The equation is true if both sides are equal.

Step 3: Check Pair (0,2)

For pair (0,2), substitute y=2 into the equation. This gives \(2+2=0\) which simplifies to 4=0. This is false so the pair (0,2) is not a solution.

Step 4: Check Pair (2,0)

For pair (2,0), substitute y=0 into the equation. This gives \(0+2=0\) which simplifies to 2=0. This is false so the pair (2,0) is not a solution.

Step 5: Check Pair (0,-2)

For pair (0,-2), substitute y=-2 into the equation. This gives \(-2+2=0\) which simplifies to 0=0. This is true so the pair (0,-2) is a solution to the equation.

Key Concepts

Linear Equations

A linear equation is a type of algebraic equation that represents a straight line when plotted on a graph. In its simplest form, it can be written as ax + by = c, where a, b, and c are constants and x and y are the variables. These equations are fundamental in algebra and appear across various disciplines.

For instance, in the equation y + 2 = 0 given in the exercise, it represents a horizontal line on the coordinate plane. The y value for any point on this line will always satisfy this equation, so any point with a y other than -2 will not lie on this line, thus not satisfying the equation.

Moreover, linear equations have the property that each variable is only to the power of one, and the graph of such equations results in a straight line. Understanding the graph can also help visualize why certain ordered pairs do or do not satisfy the equation.

Substituting Values

Substituting values refers to the process of replacing a variable with a specified value to simplify an equation or function. It's an essential skill in algebra as it allows us to understand how changing one component of an equation will affect the overall outcome. In essence, substitution helps in evaluating the equation for specific cases.

When we look at the initial exercise, for each ordered pair given, we carried out substitution by taking the y value from the pair and placing it in the equation. By substituting these values, we could concretely determine whether the equality holds for each case. For example, by replacing y with 2 from the ordered pair (0,2), we obtained the equation 2 + 2 = 0, which we can see is not true.

Effective substitution can only occur if one executes the replacement accurately and abides by the rules of arithmetic and algebraic operations, as a small error can lead to an incorrect conclusion about the solution to the equation.

Solving Algebraic Equations

Solving algebraic equations is the process of finding the value(s) for variable(s) that make an equation true. This involves a variety of techniques such as simplifying expressions, isolating variables, and using operations that reverse one another, such as addition and subtraction or multiplication and division.

In the context of our exercise, solving the equation y + 2 = 0 is straightforward since it's already simplified with only one variable. The solution involves isolating y by performing the operation opposite of addition which is subtraction. When we subtract 2 from both sides of the equation, we find that y = -2. Hence, any ordered pair where y is not equal to -2 will not solve the equation.

Understanding how to solve these simple algebraic equations can build a strong foundation for tackling more complicated equations that involve multiple variables and more complex operations.