Problem 21 - Algebra 1

Question

Find three different ordered pairs that are solutions of the equation. \(y=3 x-5\)

Step-by-Step Solution

Answer

The solution to this exercise are the following ordered pairs: (1, -2), (2, 1) and (0, -5). These are three different pairs that are solutions to the equation \(y=3x-5\).

Step 1: Substitution for the first pair

Choose a value for \(x\). Let's start by letting \(x = 1\). Substitute \(1\) for \(x\) in the equation and solve for \(y\):\n\[(y = 3(1) - 5) => (y = 3 - 5) => (y = -2)\]\nSo, (1, -2) is the first ordered pair that is a solution to the equation.

Step 2: Substitution for the second pair

Choose a different value for \(x\). Let's let \(x = 2\). Substitute \(2\) for \(x\) in the equation and solve for \(y\):\n\[(y = 3(2) - 5) => (y = 6 - 5) => (y = 1)\]\nSo, (2, 1) is the second ordered pair that is a solution to the equation.

Step 3: Substitution for the third pair

Choose a third value for \(x\). This time let's choose \(x = 0\). Substitute \(0\) for \(x\) in the equation and solve for \(y\):\n\[(y = 3(0) - 5) => (y = 0 - 5) => (y = -5)\]\nSo, (0, -5) is the third ordered pair that is a solution to the equation.

Key Concepts

Ordered Pairs

Ordered pairs are fundamental elements in the study of algebra, and specifically in coordinate geometry. They are written in the form \( (x, y) \), representing points on a two-dimensional plane, where \( x \) corresponds to the horizontal value (abscissa), and \( y \) corresponds to the vertical value (ordinate). These pairs indicate the position of points relative to two intersecting lines, usually referred to as axes.

When we are trying to find ordered pairs that satisfy a particular algebraic equation, we are essentially looking for points that lie on the graph of that equation. In the equation \( y = 3x - 5 \), any pair that makes the equation true when you substitute \( x \) and \( y \) with their values is a part of the solution set. These pairs are useful for graphing the equation and for understanding the relationship between the variables involved.

Substitution Method

The substitution method is a powerful technique used to solve algebraic equations, particularly handy when working with systems of equations. However, it can also be applied to find individual solutions to single equations, as seen in the example exercise.

In the substitution method, we choose a value for one variable and substitute that value into the equation to solve for the other variable. This method simplifies the process of finding solutions. In the given exercise, when \( x \) was set to a specific value, that number was substituted in place of \( x \) in the equation, which then allowed for solving for \( y \). It's like unlocking a combination lock: once you know one number, the rest becomes easier to figure out.

The substitution method is reliable and straightforward, making it an excellent technique for students to master as they progress in algebra. By picking different values for \( x \) and employing substitution, various ordered pairs were discovered that satisfy the given linear equation.

Linear Equations

Linear equations form the backbone of algebra and are identifiable by their straight-line graphs. These equations represent constant relationships between two variables and are generally written in the form \( y = mx + b \), where \( m \) denotes the slope of the line—the steepness or incline—and \( b \) indicates the y-intercept, the point where the line crosses the y-axis.

One of the key characteristics of linear equations is that their graphs are straight lines. This makes finding solutions relatively straightforward, as any point lying on the line is a solution to the equation. Therefore, identifying different points, or ordered pairs, often involves choosing values for one variable (like \( x \) in the exercise) and solving for the other (\( y\)).

A thorough understanding of linear equations is essential since they are not only a foundational concept in algebra but also have broad applications across mathematics and various scientific fields, from predicting trends to formulating hypotheses in experimental studies.