Problem 31 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ y=3 x-5 $$

Step-by-Step Solution

Answer

The ordered pairs (-1, -8), (0, -5), and (1, -2) are the solutions of the given equation.

Step 1: Understand the Question

The exercise is asking for three ordered pairs that satisfy the equation y = 3x - 5. An ordered pair is a pair of numbers (x, y) where 'x' is the first element and 'y' is the second.

Step 2: Choose arbitrary x-values

Choose three different numbers for 'x'. For example, x = -1, x = 0 and x= 1, can be chosen.

Step 3: Substitute x-values in the equation

Substitute these x values in the equation of the line one by one, then calculate the corresponding y values. For instance, substituting x=-1 in y = 3x -5 gives \(y = 3*(-1) -5 = -8\), substituting x = 0 gives \(y = 3*0 -5 = -5\) and substituting x = 1 gives \(y = 3*1 - 5 = -2\).

Step 4: Form the ordered pairs

The ordered pairs are formed by combining each x-value with its corresponding y-value. Thus, the three ordered pairs that satisfy the equation are (-1, -8), (0, -5) and (1, -2).

Key Concepts

Ordered Pairs

The concept of ordered pairs is foundational in mathematics and helps us represent points on the coordinate plane. An ordered pair consists of two elements written in a specific order, usually as

• First element: the x-coordinate (horizontal position), also known as the abscissa
• Second element: the y-coordinate (vertical position), also known as the ordinate

For example, in the ordered pair (2, 3), '2' represents the x-coordinate, and '3' represents the y-coordinate. The order is always important because (2, 3) is not the same as (3, 2). You often find ordered pairs in problems involving linear equations, and they are essential for identifying points on a graph. Remember that each ordered pair can be plotted as a specific dot on the coordinate plane, giving you a visual representation of the solution of an equation.

Substitution Method

The substitution method, often used to solve linear equations, is a straightforward technique that allows us to find unknown values. It involves replacing a variable with a numerical value or another expression. In this method, we

• Select a value to substitute for the variable
• Replace the variable in the equation with the chosen value
• Calculate the result to find the other unknown

For the equation given in the exercise, by substituting specific x-values into the equation \(y = 3x - 5\), we find corresponding y-values. This step-by-step approach makes it clear how the equation behaves. By calculating several ordered pairs, such as those where x is -1, 0, and 1, we discover specific solutions that fit within the equation. This technique is particularly helpful in verifying solutions and understanding the relationship between x and y in linear equations.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by the intersection of two number lines: the x-axis and the y-axis. This plane is pivotal in graphing and understanding ordered pairs visually.

• The x-axis runs horizontally, and values can be positive or negative.
• The y-axis runs vertically, with values also being positive or negative.
• The point where both axes meet is called the origin, marked as (0,0).

When solving the linear equation \(y = 3x - 5\), we use the coordinate plane to plot the ordered pairs we found as solutions. Each pair, like (-1, -8), (0, -5), and (1, -2), corresponds to a unique point on this plane. By connecting these points, we can visualize the line described by the equation. Understanding this layout helps us see not only where points lie but also the linear relationship demonstrated in the equation. Linear equations on the plane are typically straight lines, showing a constant rate of change between x and y.