Problem 34 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ x+2 y=8 $$

Step-by-Step Solution

Answer

Therefore, the three ordered pairs that are solutions of the equation \(x + 2y = 8\) are (0, 4), (2, 3), and (4, 2).

Step 1: Understand the equation

The equation to solve is \(x + 2y = 8\). Here, x and y are variables that can take any values such that they satisfy the equation.

Step 2: Choose a value for x and solve for y

Choose x = 0 and substitute it into the equation. Then solve for y. So, \(0 + 2y = 8\), which simplifies to \(2y = 8\) and further simplifies to y = 8/2 = 4. Thus, when x = 0, y = 4.

Step 3: Choose another value for x and solve for y

Now choose x = 2 and substitute it into the equation. Then solve for y. Hence, \(2 + 2y = 8\), which simplifies to \(2y = 8 - 2 = 6\) and further simplifies to y = 6/2 = 3. So, when x = 2, y = 3.

Step 4: Choose a third value for x and solve for y

Next, choose x = 4 and substitute it back into the equation. Solve for y: \(4 + 2y = 8\), this simplifies to \(2y = 8 - 4 = 4\), and further simplifies to y = 4/2 = 2. Hence, when x = 4, y = 2.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to define the position of a point in a coordinate system. An ordered pair is written in the form **(x, y)**, where **x** is the horizontal coordinate and **y** is the vertical coordinate. The order of the numbers is crucial, as changing them changes the position they represent in a graph.

In the context of solving linear equations, ordered pairs help us identify specific solutions. Each pair corresponds to a point on the line represented by the equation.

• The first value of the pair corresponds to the variable **x**.
• The second value corresponds to the variable **y**, determined based on the equation.

For the linear equation given, such as \(x + 2y = 8\), finding an ordered pair means selecting a value for **x** and calculating the corresponding **y** value to satisfy the equation. For example, choosing \(x = 0\) results in the ordered pair (0, 4), which is one solution to the equation.

Solution of Equations

A solution of an equation is any set of values for the variables involved that make the equation true. In the case of linear equations like \(x + 2y = 8\), solutions are ordered pairs of numbers **(x, y)** that fit the equation precisely.

To find solutions, you follow a simple procedure:

• Choose a value for one variable (typically **x**).
• Substitute this value into the equation.
• Solve the resulting equation for the other variable (typically **y**).

When repeating these steps with different values for **x**, we can find multiple solutions. These solutions form a straight line on a graph, demonstrating that linear equations have infinitely many solutions. This exercise illustrated this by solving for **y** with different **x** values: 0, 2, and 4, each giving a unique ordered pair.

Variables

Variables are symbols used in mathematics to represent numbers whose values can change. They are fundamental in forming equations and mathematical expressions. In the equation \(x + 2y = 8\), **x** and **y** are variables:

• **x** is the independent variable.
• **y** is the dependent variable because its value depends on what you substitute for **x**.

Choosing different values for **x** helps determine different corresponding values for **y**. This dynamic interaction portrays how a single equation can express an infinite number of solutions. Understanding variables is vital in mathematics as it allows generalization and flexibility in creating and solving complex problems.