Problem 19 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$x-y=5$$

Step-by-Step Solution

Answer

Three solutions are \((0, -5), (5, 0), (10, 5)\). Plot and connect these points to draw the graph.

Step 1: Rearrange the Equation

First, rearrange the equation to the slope-intercept form, which is \( y = mx + c \). The given equation is \( x - y = 5 \). Add \( y \) to both sides and subtract 5 from both sides to get \( y = x - 5 \).

Step 2: Choose Values for x

We will find three solutions by selecting three different values for \( x \) and then finding the corresponding \( y \) values. We will choose \( x = 0, 5, \, \text{and} \, 10 \).

Step 3: Calculate y for x=0

Substitute \( x = 0 \) into the equation \( y = x - 5 \). This gives \( y = 0 - 5 = -5 \). So, one solution is \((0, -5)\).

Step 4: Calculate y for x=5

Substitute \( x = 5 \) into the equation \( y = x - 5 \). This gives \( y = 5 - 5 = 0 \). So, another solution is \((5, 0)\).

Step 5: Calculate y for x=10

Substitute \( x = 10 \) into the equation \( y = x - 5 \). This gives \( y = 10 - 5 = 5 \). So, a third solution is \((10, 5)\).

Step 6: Plot the Points on a Graph

Now, plot the three points \((0, -5), (5, 0), \text{and} (10, 5)\) on a graph. Draw a straight line through these points to represent the equation \( y = x - 5 \). This line shows all possible solutions to the equation.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing and analyzing lines. This form is written as \( y = mx + c \), where:

• \( y \) represents the dependent variable.
• \( m \) stands for the slope of the line, indicating its steepness.
• \( x \) is the independent variable.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, isolate \( y \) on one side of the equation. For example, the equation \( x - y = 5 \) can be rearranged by adding \( y \) to both sides and subtracting 5, resulting in \( y = x - 5 \). This is now in slope-intercept form, making it clear that the slope \( m \) is 1, and the y-intercept \( c \) is -5. Understanding this form simplifies graphing as you can instantly see the direction and position of the line on a graph.

Coordinate System

The coordinate system is a way of mapping data into a visual format. It helps us to easily understand relationships between variables through points and lines.
It consists mainly of two axes:

• The horizontal axis known as the x-axis.
• The vertical axis known as the y-axis.

These axes intersect at a point called the origin, labeled as (0, 0). Each point on this plane can be represented by an ordered pair \((x, y)\), where \(x\) indicates the position on the x-axis and \(y\) indicates the position on the y-axis.
In our exercise, once we have derived the equation of the line in the slope-intercept form \( y = x - 5 \), finding solutions involves selecting values for \( x \) and calculating the corresponding \( y \) values. The pairs \((x, y)\) are what you will plot on the coordinate system, and connecting these will give a visual representation of the equation.

Solving for Y

Solving for \( y \) means making \( y \) the subject of the formula, which allows you to find its value for any given \( x \). In our example, to rearrange the equation \( x - y = 5 \), we add \( y \) to both sides, giving us \( y = x - 5 \).
With this equation, you can substitute different \( x \) values to find matching \( y \) values:

• When \( x = 0 \), \( y = 0 - 5 = -5 \). So, the point is (0, -5).
• When \( x = 5 \), \( y = 5 - 5 = 0 \). So, the point is (5, 0).
• When \( x = 10 \), \( y = 10 - 5 = 5 \). So, the point is (10, 5).

These solutions give you specific points that lie on the line. Plotting these points and drawing a line through them visually demonstrates the relationship between \( x \) and \( y \) for the given equation. Each point on this line is a solution to the equation.