Problem 36 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$y=-3 x \quad(0,0),(-3,0),(-1,3)$$

Step-by-Step Solution

Answer

(0,0) and (-1,3) are solutions; (-3,0) is not.

Step 1: Understand the Equation

The given equation is a linear equation in the form of \(y = -3x\). This means for any point \((x, y)\) to be a solution, it must satisfy the equation when the \(x\) value of the point is substituted into the equation to find the \(y\) value.

Step 2: Test the First Point

Substitute \(x = 0\) from the point \((0,0)\) into the equation \(y = -3x\). This gives \(y = -3(0) = 0\). The y-value is 0, and since this matches the point \((0,0)\), it is a solution.

Step 3: Test the Second Point

Substitute \(x = -3\) from the point \((-3,0)\) into the equation \(y = -3x\). This gives \(y = -3(-3) = 9\). The y-value here is 9, which does not match the point \((-3,0)\). Therefore, it is not a solution.

Step 4: Test the Third Point

Substitute \(x = -1\) from the point \((-1,3)\) into the equation \(y = -3x\). This gives \(y = -3(-1) = 3\). The y-value is 3, and this matches the point \((-1,3)\), so it is a solution.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in coordinate geometry and algebra. They are typically written as \((x, y)\), where \(x\) is the horizontal component and \(y\) is the vertical component. These pairs represent points on a two-dimensional graph.

Key Characteristics of Ordered Pairs:

• The order matters: The point \((x, y)\) is different from \((y, x)\).
• Used to define the location of a point in a coordinate system.
• Applicable in various mathematical equations to find or verify solutions.

In the context of linear equations, checking ordered pairs helps determine if they lie on the line described by the equation.Understanding ordered pairs is crucial as they provide a basis for graphing and solving equations.

Solutions

In the realm of linear equations, a solution is a set of values that satisfy the equation. When dealing with ordered pairs, a solution means that, upon substitution of the \(x\) value into the equation, the resulting \(y\) value corresponds to the given \(y\) in the pair, making the ordered pair a 'solution' to the equation.

How to Verify Solutions for Linear Equations:

• Substitute the \(x\) value of the ordered pair into the equation.
• Calculate the resulting \(y\) value.
• Compare it with the \(y\) value in the ordered pair.
• If they match, the ordered pair is a solution.

A solution confirms that the point lies on the line represented by the equation. In our example, \((0,0)\) and \((-1,3)\) are solutions to the equation \(y = -3x\).

Substitution Method

The substitution method is a powerful tool often used in algebra to solve equations, including verifying solutionsfor ordered pairs. This technique involves substituting values from the ordered pair into the equation to check if it holds true.

Steps Involved in the Substitution Method:

• Identify the \(x\) value from the ordered pair.
• Replace \(x\) in the equation with the identified value.
• Perform the necessary calculations to find \(y\).
• Verify if the calculated \(y\) value matches the \(y\) in the ordered pair.

This method is particularly helpful as a straightforward and systematic approach to determine whether an ordered pairis a solution to the equation.In our exercise, we applied this method to each ordered pair to establish whether they satisfied the equation \(y = -3x\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry by using an ordered pair systemto describe the location of points. It plays a critical role in working with linear equations and understanding howthey plot on a graph.

Applications of Coordinate Geometry:

• Graphing lines and identifying slopes and intercepts.
• Locating points using ordered pairs.
• Determining if points lie on a specific line through solving the line equation for the ordered pair values.

Understanding coordinate geometry is essential when dealing with linear equations, as it provides a visual representationof the solutions and helps reinforce the relationships between algebraic properties and geometric visualization.For example, the equation \(y = -3x\) represents a straight line on a graph, where the ordered pairs that satisfy theequation lie directly on this line.