Problem 14 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Yes, the ordered pair (-2, -2) is a solution to the equation \(y=-2\).

Step 1: Identify the 'y' in equation and from ordered pair

In the equation \(y = -2\), 'y' is -2. According to the given ordered pair (-2, -2), 'y' is also -2.

Step 2: Compare

Now, let's compare these values. If they are equal, then the ordered pair is indeed a solution to the equation.

Step 3: Verify

In this case, both values of 'y' are -2; therefore, the ordered pair (-2, -2) is a solution of the equation \(y= -2\).

Key Concepts

Ordered Pairs

An ordered pair is simply a pair of numbers usually written in the form \(x, y\). It represents a point on a coordinate plane. The first element, or coordinate of the pair, is the 'x' value, and the second element is the 'y' value. Ordered pairs are crucial in identifying specific points. For example, the ordered pair \((-2, -2)\) specifies a point where both the 'x' and 'y' coordinates have the value -2.
Sometimes, ordered pairs can also be viewed in functions or equations to check if they satisfy them. When we input the 'x' value into an equation and obtain the corresponding 'y' value, these values combine to form an ordered pair.
To quickly recap:

• An ordered pair is displayed as \(x, y\).
• The order is essential; switching the position of numbers changes the meaning.
• Ordered pairs are used to determine specific points or solutions.

Solutions of Equations

A solution of an equation is a value or set of values that satisfies the equation. When dealing with linear equations, such as \(y = -2\), a solution can be verified by substituting the values from an ordered pair into the equation. If the equation holds true, then this pair is a solution.
For example, given the equation \(y = -2\) and the ordered pair \((-2, -2)\), we substitute the 'y' value \(-2\) from the ordered pair into the equation. Since it matches the equation's 'y' value, the ordered pair is a solution. This tells us that the point lies on the line represented by the equation.
Key points to remember regarding solutions:

• A solution fulfills the condition set by the equation.
• Substituting the ordered pair into the equation must make the equation true.
• In coordinate form, both x and y must satisfy the equation for it to be a solution.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane defined by a horizontal line (x-axis) and a vertical line (y-axis). Each point on this plane is unique, represented by an ordered pair \(x, y\).
The intersection point of the x-axis and the y-axis is referred to as the origin, designated as the point \(0, 0\). Moving horizontally or vertically from the origin allows locating other points, determined by their specific ordered pairs.
When representing equations or solutions on a coordinate plane, these ordered pairs show exact locations where the conditions of the equation hold true. This visualization aids in understanding the relationship between 'x' and 'y' values. For instance, for the equation \(y = -2\), every point on a line parallel to the x-axis where 'y' equals -2 is a solution.
A quick overview of the coordinate plane:

• It's composed of two axes: x (horizontal) and y (vertical).
• The origin is the center point, \(0, 0\).
• Ordered pairs identify points on this plane accurately.
• Visualizing solutions and equations on the plane enhances comprehension.