Problem 69 - Algebra A Combined Function

Question

Complete each ordered pair for the given equation. $$ y=4.2 x ;(-2, \quad) $$

Step-by-Step Solution

Answer

The complete ordered pair is \((-2, -8.4)\).

Step 1: Identify the Given Equation

The equation provided is \( y = 4.2x \). This equation shows the relationship between the variables \( x \) and \( y \).

Step 2: Substitute the Value of x

You need to find \( y \) when \( x = -2 \). Substitute \( x = -2 \) into the equation: \( y = 4.2(-2) \).

Step 3: Perform the Calculation

Multiply 4.2 by -2 to find \( y \): \( y = 4.2 \times (-2) = -8.4 \).

Step 4: Complete the Ordered Pair

Now that you have calculated \( y \), the ordered pair becomes \((-2, -8.4)\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics, particularly in the context of graphing and coordinate systems. It consists of two elements: an "x" value and a "y" value. These values are contained within parentheses and are separated by a comma. The first value corresponds to the horizontal position (x-coordinate), while the second value represents the vertical position (y-coordinate). Together, an ordered pair like \((-2, y)\) can represent a point on a two-dimensional plane.

In this exercise, ordered pairs are used to express solutions to an equation. When completing an ordered pair, our goal is to determine the unknown value using the equation provided.

The process involves substituting the given x-value into the equation to find the corresponding y-value, which completes the ordered pair. Thus, handling ordered pairs involves both understanding their structure and computing values to create complete, meaningful pairs.

Variable Substitution

Variable substitution is a powerful technique used to solve equations by replacing one variable with a given numerical value. In simple terms, you take a known value for the variable and plug it into the equation. This allows you to solve for another unknown variable by performing standard arithmetic calculations.

In the example problem, you start with the equation \( y = 4.2x \). You are tasked with finding the corresponding "y" value when \( x = -2 \). Using substitution, you replace "x" with "-2" in the equation, which modifies it to \( y = 4.2(-2) \). This transformation is crucial for simplifying the problem, enabling you to solve for "y."

Through substitution, complex problems are broken down into simpler calculations, providing clarity and direct pathways to solutions. It is an essential problem-solving tool, especially in algebra, as it bridges given information to the sought-after results.

Basic Arithmetic Operations

Arithmetic operations are the building blocks of mathematics, involving basic calculations such as addition, subtraction, multiplication, and division. In our example, we primarily focus on multiplication. When we substitute \( x = -2 \) into the equation \( y = 4.2x \), it becomes an arithmetic operation: \( 4.2 \times (-2) \).

This multiplication is straightforward: you multiply the number 4.2 by -2. Upon solving, you obtain -8.4 for "y."

Multiplication and other basic operations follow straightforward rules and are essential for finding precise solutions to equations. Mastering these operations allows you to solve more complicated equations as well. Keep practicing, and soon these operations will become second nature in tackling mathematical problems.