Problem 13 - Prealgebra

Question

For each equation, complete the given ordered pairs. $$y=-2 x \quad(0, \quad),(-2,),(2,)$$

Step-by-Step Solution

Answer

The complete ordered pairs are \((0, 0)\), \((-2, 4)\), and \((2, -4)\).

Step 1: Understanding the Equation

The given equation is a linear equation in the form of \(y = -2x\). This equation tells us that the value of \(y\) is equal to \(-2\) times the value of \(x\). We will use this equation to find the missing \(y\)-values for the given ordered pairs.

Step 2: Completing the First Ordered Pair

For the first ordered pair \((0, \quad)\), substitute \(x = 0\) into the equation:\[y = -2(0)\]Calculate the value of \(y\):\[y = 0\]Thus, the complete ordered pair is \((0, 0)\).

Step 3: Completing the Second Ordered Pair

For the second ordered pair \((-2, \quad)\), substitute \(x = -2\) into the equation:\[y = -2(-2)\]Calculate the value of \(y\):\[y = 4\]Thus, the complete ordered pair is \((-2, 4)\).

Step 4: Completing the Third Ordered Pair

For the third ordered pair \((2, \quad)\), substitute \(x = 2\) into the equation:\[y = -2(2)\]Calculate the value of \(y\):\[y = -4\]Thus, the complete ordered pair is \((2, -4)\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics, especially within coordinate geometry. It consists of two elements written in a specific sequence, typically denoted as \((x, y)\). The first element represents the horizontal coordinate (x-axis), while the second element represents the vertical coordinate (y-axis). This structure allows for precise locations to be pinpointed on a two-dimensional graph.

When dealing with linear equations, ordered pairs are useful in representing solutions. For instance, if a linear equation relates \(x\) and \(y\), any number pair that satisfies the equation is an ordered pair. For example, in the equation \(y = -2x\), the ordered pair \(0, 0\) tells us that when \(x = 0\), \(y = 0\).

Ordered pairs can illustrate trends and relationships in data when plotted on a coordinate plane, making them valuable tools in mathematical analysis.

Substitution Method

The substitution method is a key technique used to solve equations, particularly useful for finding unknown values within a system of equations. This method involves replacing a variable in one equation with another expression to simplify the process of finding the solution.

To use the substitution method with linear equations:

• Identify a variable to express in terms of another.
• Replace this variable in the second equation with the expression found.
• Solve the resulting equation for the remaining variable.

In the original exercise's context, the substitution method was utilized by inserting the values of \(x\) into the equation \(y = -2x\) to find the corresponding \(y\) values, thus completing the ordered pairs. For example, substituting \(x = -2\) into the equation gives \(y = 4\), resulting in the ordered pair \(-2, 4\).

This approach efficiently deduces solutions by simplifying the system into a single equation that is easier to manage.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra with geometry to solve problems concerning distance, angles, and more on a coordinate plane. It involves plotting points, lines, and equations on graphs to visually interpret algebraic relationships.

In the context of the given exercise, coordinate geometry is applied to interpret the linear equation \(y = -2x\). This equation can be visualized as a straight line on the coordinate plane, where every point (or ordered pair) on the line satisfies the equation.

The equation \(y = -2x\) demonstrates a straight line with a negative slope of \(-2\). The slope indicates a downward trend, meaning for every unit increase in \(x\), \(y\) decreases by 2 units. The line passes through the origin (0, 0) since when \(x = 0\), \(y\) also equals 0.

Understanding how linear equations translate into geometric lines connects algebraic solutions with tangible geometric objects, offering deeper insight into the nature of linear relationships.