Problem 21 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ y=-5 $$

Step-by-Step Solution

Answer

The three ordered pairs that are solutions of the equation \( y=-5 \) are (-2,-5), (0,-5) and (2,-5).

Step 1: Understand the given equation

The given equation is \( y=-5 \). This is a horizontal line that passes through the point (0, -5) on the number line. Therefore, every point on this line should have the y-coordinate of -5.

Step 2: Choose values for x

Since the value of 'y' is fixed to -5, it allows 'x' to take on any real number. We can choose three different values for 'x'. For example, let’s choose -2, 0, and 2.

Step 3: Formulate ordered pairs

Substitute the chosen 'x' values into the equation to get the 'y' values, though in this case, for all 'x' values, the 'y' value will be -5. Therefore, the ordered pairs solutions to the equation \( y = -5 \) are (-2,-5), (0,-5), and (2,-5).

Key Concepts

Ordered Pairs

An ordered pair consists of two numbers written in a specific order, usually as \(x, y\). The order in which these numbers appear matters, and they represent a position in the coordinate plane where 'x' is the horizontal position and 'y' is the vertical position.

Ordered pairs are critical in representing solutions to equations. For the equation \( y = -5 \), it means that, irrespective of the value of 'x', 'y' will always be -5.

• This results in ordered pairs like (-2, -5), (0, -5), and (2, -5).
• The first number in each pair can vary, while the second number remains constant.

Ordered pairs are a fundamental part of coordinate geometry, helping define exact locations on a graph.

Solution of Equations

A solution of an equation is a set of values that, when substituted into the equation, make the equation true. For equations with two variables such as \( y = -5 \), solutions are often represented as ordered pairs.

In this specific example, because the equation \( y = -5 \) only assigns a value to 'y', 'x' can be any real number, leading to infinite possible solutions in the form of \(x, -5\).

• The equation describes a horizontal line where every point has a 'y' value of -5.
• By choosing different x-values like -2, 0, and 2, we determine the ordered pairs (-2, -5), (0, -5), and (2, -5), which are all valid solutions.

Thus, understanding how an equation describes geometric representations helps in determining the solutions visually on a graph.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves that represent solutions of equations. It's divided into four quadrants by the x-axis (horizontal line) and y-axis (vertical line).

Points on this plane are represented as ordered pairs \(x, y\), where 'x' and 'y' are numbers that describe the position of the point.

• For instance, the equation \( y = -5 \) is graphed as a horizontal line that crosses the y-axis at -5.
• All points on this line like (-2, -5), (0, -5), and (2, -5) signify this same horizontal level.

Each point gives us a visual representation of the solutions, demonstrating how equations describe specific locations and shapes on the plane. Understanding this helps us visualize math in a physical space, making it more intuitive and engaging.