Problem 32 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$y=-3 x+2 \quad(0,-3),(0,2),(-3,0)$$

Step-by-Step Solution

Answer

The ordered pair (0, 2) is the solution for the equation.

Step 1: Understand the Equation

The given equation is \( y = -3x + 2 \), which represents a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to determine whether each ordered pair satisfies this equation.

Step 2: Analyze the Ordered Pair (0, -3)

Substitute \( x = 0 \) and \( y = -3 \) into the equation: \( -3 = -3(0) + 2 \). Simplifying gives \( -3 = 2 \), which is false. Therefore, the pair (0, -3) is not a solution.

Step 3: Analyze the Ordered Pair (0, 2)

Substitute \( x = 0 \) and \( y = 2 \) into the equation: \( 2 = -3(0) + 2 \). Simplifying gives \( 2 = 2 \), which is true. Therefore, the pair (0, 2) is a solution.

Step 4: Analyze the Ordered Pair (-3, 0)

Substitute \( x = -3 \) and \( y = 0 \) into the equation: \( 0 = -3(-3) + 2 \). Simplifying gives \( 0 = 9 + 2 \) or \( 0 = 11 \), which is false. Therefore, the pair (-3, 0) is not a solution.

Step 5: Determine the Valid Solutions

Based on the calculations, (0, 2) is the only ordered pair that satisfies the equation \( y = -3x + 2 \).

Key Concepts

Slope-Intercept Form

Linear equations are often represented in what is known as the slope-intercept form. This form is written as \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

For our specific equation \( y = -3x + 2 \), the slope \( m \) is \(-3\), meaning the line descends or moves downward by 3 units for every 1 unit it moves to the right. The y-intercept \( b \) is 2, so the line crosses the y-axis at the point (0, 2).
The slope-intercept form gives us a clear picture of how the line behaves on a graph without plotting multiple points. By knowing just the slope and the y-intercept, we can quickly draw and understand the line's position and direction.

Understanding Ordered Pairs

Ordered pairs are crucial in determining the location of points on a graph. Each ordered pair consists of an \( x \)-coordinate and a \( y \)-coordinate, represented in the format \((x, y)\).
In the exercise, we are tasked with finding out if given ordered pairs are solutions to the linear equation \( y = -3x + 2 \). This means we're checking if when we plug in these \( x \) and \( y \) values into the equation, both sides of the equation are equal or true.

• For example, with the pair (0, 2): substituting \( x = 0\) and \( y = 2\) verifies the equation because \(2 = 2\).
• However, with the pair (0, -3), substituting \( x = 0\) and \( y = -3\) gives \(-3 = 2\), which is incorrect.

Evaluating ordered pairs in this manner helps confirm which points lie on the graph of the equation.

Solution Verification

Solution verification is the process of confirming that a particular ordered pair satisfies the given equation. It involves substituting the \( x \) and \( y \) values from the pair into the equation and checking if the equation holds true.
For the linear equation \( y = -3x + 2 \), let's see how verification works:

• If substituting an ordered pair results in both sides of the equation being equal, the pair is a valid solution, meaning it lies on the line described by the equation.
• If the results do not match, the ordered pair is not a solution and does not lie on the line.

In our task, (0, 2) verifies as a solution because substituting \( x \) and \( y \) gives \( 2 = 2 \). The others do not verify because when substituted, the equalities do not hold true. Solution verification is a key step in problems involving graphing and understanding linear equations, ensuring accuracy and deeper insight into the equation's nature.