Problem 18 - Prealgebra

Question

For each equation, complete the given ordered pairs. $$y=-\frac{1}{3} x+1 \quad(-3,1,(0,),(3,)$$

Step-by-Step Solution

Answer

The complete ordered pairs are (0, 1) and (3, 0).

Step 1: Understand the Equation

The equation given is a linear equation in the form of \( y = mx + b \) where \( m = -\frac{1}{3} \) is the slope and \( b = 1 \) is the y-intercept. This means for any value of \( x \), you can find \( y \) by substituting \( x \) into the equation.

Step 2: Calculate for (0, _)

To find the corresponding \( y \)-value for \( x = 0 \), substitute \( x = 0 \) into the equation: \[ y = -\frac{1}{3}(0) + 1 \]Thus, \( y = 1 \). So the complete ordered pair is (0, 1).

Step 3: Calculate for (3, _)

Next, substitute \( x = 3 \) into the equation to find the \( y \)-value:\[ y = -\frac{1}{3}(3) + 1 \]\[ y = -1 + 1 \]Thus, \( y = 0 \). The complete ordered pair is (3, 0).

Key Concepts

Understanding the Slope-Intercept Form

Linear equations are fundamental in algebra, often expressed in the form \( y = mx + b \), known as the slope-intercept form. This form makes it straightforward to identify the slope and y-intercept of a line.

• The variable \( m \) represents the slope of the line. Slope indicates how much \( y \) changes for a unit change in \( x \). In the given equation, the slope \( m \) is \(-\frac{1}{3}\), meaning for each increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{3} \).
• The variable \( b \) is the y-intercept, which is the point where the line crosses the y-axis. In our problem, \( b \) is 1, indicating that the line cuts through the y-axis at the point 0,1.

Grasping the slope-intercept form allows you to easily predict how changes in \( x \) will affect \( y \) and gives a clear view of the line's behavior.

What are Ordered Pairs?

Ordered pairs are a fundamental concept in coordinate geometry, used to describe the location of points on a graph. Each pair consists of an \( x \)-coordinate and a \( y \)-coordinate, generally written as \( (x, y) \).

• The first number in the pair is the \( x \)-coordinate, which tells us how far left or right the point is from the origin.
• The second number is the \( y \)-coordinate, indicating how far up or down the point is from the origin.

In the equation \( y = -\frac{1}{3}x + 1 \), we have determined pairs (-3, 2), (0, 1), and (3, 0). These pairs can be plotted on a graph to show the location of points along the line described by this equation.

Applying the Substitution Method

The substitution method is a handy tool in solving equations and evaluating expressions. In the context of linear equations, it involves replacing a variable with a given value to solve for another variable.
To find the missing \( y \)-values for the given \( x \)-coordinates in ordered pairs, you substitute the \( x \)-value into the equation and solve for \( y \).

• For instance, when \( x = 0 \), replace \( x \) with 0 in the equation: \( y = -\frac{1}{3}(0) + 1 \). This simplifies to \( y = 1 \), thus completing the ordered pair as (0, 1).
• Similarly, substitute \( x = 3 \): \( y = -\frac{1}{3}(3) + 1 \). Calculating this gives \( y = 0 \), so the ordered pair is (3, 0).

This method ensures precise calculation of missing variables within ordered pairs, crucial for accurately representing relationships on a graph.