Problem 30 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=-\frac{1}{3} x$$

Step-by-Step Solution

Answer

The three solutions are (-3, 1), (0, 0), and (3, -1).

Step 1: Choose Values for x

To find three solutions, start by selecting three different values for \(x\). We'll use \(x = -3\), \(x = 0\), and \(x = 3\) for simplicity.

Step 2: Calculate y for x = -3

Use the equation \(y = -\frac{1}{3}x\). Substitute \(x = -3\): \[y = -\frac{1}{3}(-3) = 1\]. So, one point is \((-3, 1)\).

Step 3: Calculate y for x = 0

Substitute \(x = 0\) into the equation: \[y = -\frac{1}{3}(0) = 0\]. So, another point is \((0, 0)\).

Step 4: Calculate y for x = 3

Substitute \(x = 3\) into the equation: \[y = -\frac{1}{3}(3) = -1\]. Thus, the third point is \((3, -1)\).

Step 5: Plotting Points on the Graph

With the three points \((-3, 1), (0, 0), (3, -1)\) identified, plot them on a coordinate graph and connect them with a straight line since the equation is linear.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra, allowing us to visualize solutions on a graph. When you have a linear equation like \( y = -\frac{1}{3} x \), it represents a straight line. To graph it, you first need points, and you find these by choosing values for \( x \) and solving for \( y \). Here's what you do:

• Pick a few simple values for \( x \), such as \(-3, 0, \) and \(3\).
• Substitute each \( x \) value into the equation and solve for \( y \).
• Plot these points, like \((-3,1), (0,0), (3,-1)\), on your coordinate plane.

After you plot your points, use a ruler to draw a straight line through them. This line represents all possible solutions of the equation. Graphing visually shows the relationship between \( x \) and \( y \), helping you to understand the constant rate of change, or slope, which in this case is \(-\frac{1}{3}\).
The slope tells you that for every increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{3} \). This straightforward approach of plotting helps in grasping more complex graphing tasks later.

Coordinate Plane

The coordinate plane is the backdrop where all the action of graphing equations happens. Imagine it as a huge grid that lets you plot points and lines. Here's a quick overview:

• The plane has two axes, the horizontal \(x\)-axis and the vertical \(y\)-axis, which meet at the origin \((0,0)\).
• The plane is divided into four quadrants, numbered I to IV, which help in identifying the signs of the coordinates in each part.
• Points on this plane are indicated using coordinates in the form \((x, y)\).

For instance, the point \((3, -1)\) is located 3 units to the right of the origin and 1 unit down. Understanding the coordinate plane is crucial as it sets the stage for graphing not just lines, but also more complex figures. A well-graphed line forms a clear visual representation of the equation, showing all its solutions.
So, getting comfortable with the plane helps not only in simple plotting but also prepares for advanced graphing. Think of it as learning the alphabet before writing words.

Solutions of Equations

Solutions to equations are specific combinations of \( x \) and \( y \) that satisfy the equation, making both sides equal. When we refer to solutions of linear equations, we mean the points on the graph that lie along the line.Each solution is a pair \((x, y)\):

• The icebreaker solution for any line is the intercepts; here, \( (0, 0) \) is a clear solution since substituting 0 for \( x \) or \( y \) makes the equation true.
• Sample more by choosing different \( x \) values, solving for \( y \), producing points like \((-3, 1)\) and \((3, -1)\).

With linear equations like \( y = -\frac{1}{3} x \), every point on the line is a solution. As you graph, you see that linear equations have an infinite number of solutions that align perfectly on the drawn line. This revelation is powerful because once you plot a few points, you can draw the entire line, visually covering all solutions.
Recognizing that a line is defined entirely by its slope and a single point lets you predict and calculate other possible solutions, essential for more complex equations where visualizing all solutions quickly proves beneficial.