Problem 18 - Algebra 1

Question

Decide whether the given ordered pair is a solution of the equation. \(x=-4,(1,-4)\)

Step-by-Step Solution

Answer

No, the ordered pair (1, -4) is not a solution to the equation \(x = -4\).

Step 1: Identify the problem

The ordered pair (1, -4) is given and we're asked to check whether it is a solution to the equation \(x = -4\). The x-coordinate is the first value in the ordered pair.

Step 2: Substitute into the equation

Now, the value of x in the ordered pair, which is 1, is substituted into the equation to replace x. The equation thus becomes \(1 = -4\).

Step 3: Evaluation

The equation \(1 = -4\) is evaluated, which obviously is not true. Therefore, the provided ordered pair (1, -4) is not a solution to the equation \(x = -4\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in algebra that consists of two elements grouped together. Typically, it is represented by \(x, y\), where \x\ is the first element and \y\ is the second. It is crucial to maintain this order since swapping them would lead to a different pair altogether.

• The first element usually represents the horizontal value, commonly referred to as the x-coordinate.
• The second element represents the vertical position, known as the y-coordinate.

Ordered pairs are a key part of graphing equations since they specify exact positions on a graph. In the context of our example, the ordered pair \(1, -4\) indicates a position where \x\ is 1 and \y\ is -4. When verifying a solution for an equation, always double-check which variable the first and second elements correspond to.

Solution of Equations

Determining if an ordered pair is a solution to an equation involves substituting the corresponding values into the equation and checking if it holds true. Let's break it down further:

• Identify the variables in the equation.
• Replace each variable with the values from the ordered pair. For example, if the equation is \(x = -4\), substitute \x\ with the first value of the ordered pair.
• Evaluate to see if the equation remains true. If it does, then the ordered pair is a solution.

From our provided exercise, upon substituting \(x = 1\) into the equation \(x = -4\), we end up with \(1 = -4\), which is false. Therefore, understanding this concept can help in identifying correct solutions or even graphing points.

Coordinate System

A coordinate system is a framework that uses two perpendicular axes to help locate points on a plane. The most common type in algebra is the Cartesian coordinate system, composed of the x-axis (horizontal) and the y-axis (vertical). Each point on this system is defined by an ordered pair \(x, y\). Some important aspects of the coordinate system include:

• The origin, which is the point \(0, 0\) where the x-axis and y-axis intersect.
• The positive x-direction extends to the right, while the negative x-direction extends to the left.
• The positive y-direction extends upwards, whereas the negative y-direction goes downwards.

Understanding the coordinate system is integral for graphing equations and understanding geometric relationships. It's the backdrop against which ordered pairs are plotted and visualized, aiding in the comprehension of algebraic equations.