Problem 26 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$y=3 x-4$$

Step-by-Step Solution

Answer

Three solutions are \((0, -4)\), \((1, -1)\), and \((2, 2)\).

Step 1: Choose Values for x

Select three different values for \(x\). Common choices are \(x = 0\), \(x = 1\), and \(x = 2\).

Step 2: Calculate Corresponding y for x=0

Use the equation \(y = 3x - 4\) and substitute \(x = 0\) to calculate \(y\).\[ y = 3(0) - 4 = -4 \]So, the first point is \((0, -4)\).

Step 3: Calculate Corresponding y for x=1

Substitute \(x = 1\) into the equation to find \(y\).\[ y = 3(1) - 4 = 3 - 4 = -1 \]The second point is \((1, -1)\).

Step 4: Calculate Corresponding y for x=2

Substitute \(x = 2\) into the equation to find \(y\).\[ y = 3(2) - 4 = 6 - 4 = 2 \]The third point is \((2, 2)\).

Step 5: Plot the Points and Draw the Graph

Plot the points \((0, -4)\), \((1, -1)\), and \((2, 2)\) on the Cartesian plane. Using a ruler, draw a straight line that passes through these points to represent the graph of the equation.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental tool in graphing and understanding linear equations. Think of it as a big piece of graph paper where every position is assigned a unique pair of numbers. These numbers are called coordinates.
The coordinate plane is made up of two perpendicular lines called axes. The horizontal axis is the x-axis and the vertical axis is the y-axis.
The place where these two axes cross is called the origin, represented by the point (0, 0).

• The x-values increase as you move to the right and decrease as you move to the left on the x-axis.
• The y-values increase as you move up and decrease as you move down on the y-axis.

By plotting points on this plane, you can visualize the solution to a linear equation like a line. This helps you see how different x-values relate to y-values, and how they change together. Understanding the coordinate plane is an essential step toward mastering graphing equations.

Plotting Points

Plotting points might sound complex at first, but it's actually very simple. Each point on the graph has two numbers—an x-coordinate and a y-coordinate—that show its exact location on the coordinate plane. To plot a point, you just need to find the correct spot using these coordinates.

Here's a quick guide to plotting points:

• First, look at the x-coordinate. Move horizontally along the x-axis to find its position.
• Then, check the y-coordinate. From the x-position, move vertically to reach the y-coordinate level.
• Draw a small dot where the two coordinates intersect.

When you plot multiple points that come from an equation like \(y = 3x - 4\), they reveal the pattern or trend of the equation. For linear equations, these points will align in a straight line. This simple process creates a bridge between the abstract equation and its visual representation on the plane.

Linear Function

A linear function is one of the simplest types of functions, and it's represented graphically by a straight line. Every linear function can be expressed in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
The slope \(m\) describes how steep the line is and tells us how much \(y\) changes with a change in \(x\). For instance, in the equation \(y = 3x - 4\), the slope \(m\) is 3. This means the y-value increases by 3 for every 1 unit increase in x.
The y-intercept \(b\) is the point where the line crosses the y-axis. In \(y = 3x - 4\), the line crosses the y-axis at \(-4\).

• The linear function shows a constant rate of change, visualized as a straight line.
• The direction and steepness of the line depend on the slope.

Understanding linear functions helps in predicting and analyzing real-world situations, as many processes can be modeled using straight lines.