Problem 59 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of \(u-4 v=8\) using the horizontal axis for \(u\) values and the vertical axis for \(v\) values.

Step-by-Step Solution

Answer

Rewrite as \(v = \frac{u}{4} - 2\), plot y-intercept (-2), use slope \(\frac{1}{4}\) to find next point (4, -1), and draw the line.

Step 1: Rewrite the Equation

Rewrite the equation in a more familiar form. Start with the given equation: \[u - 4v = 8\]Isolate the variable to get it into the slope-intercept form \(y = mx + b\) (which in this case will be \(v = mu + b\)).

Step 2: Isolate the Variable

Isolate \(v\) by moving \(u\) to the other side of the equation:\[u - 4v = 8\]Subtract \(u\) from both sides:\[-4v = -u + 8\]Divide every term by -4:\[v = \frac{u}{4} - 2\]Now the equation is in slope-intercept form, \(v = \frac{u}{4} - 2\).

Step 3: Identify the Slope and Y-intercept

From the equation \(v = \frac{u}{4} - 2\), identify the slope and the y-intercept:Slope (\(m\)): \(\frac{1}{4}\)Y-intercept (\(b\)): \(-2\)

Step 4: Plot the Y-intercept

Start your graph by plotting the y-intercept of -2 on the vertical axis (where \(u = 0\)).

Step 5: Use the Slope to Find Another Point

From the y-intercept, use the slope to find another point on the graph. The slope \(\frac{1}{4}\) means you go up 1 unit and to the right 4 units. Starting from \( (0, -2)\):Move up 1 unit to -1 and then 4 units to the right to 4.You get another point (4, -1).

Step 6: Draw the Line

Draw a straight line through the points (0, -2) and (4, -1) to complete your sketch of the graph of \(u - 4v = 8\).

Key Concepts

slope-intercept form

The slope-intercept form is a way to write the equation of a line so that you can easily identify the slope and the y-intercept. The general form is \[y = mx + b\]where

• y: the dependent variable representing the vertical axis value
• x: the independent variable representing the horizontal axis value
• m: the slope of the line
• b: the y-intercept

In this form, you can easily see both the slope and the y-intercept simply by looking at the equation. For example, with the equation \[v = \frac{u}{4} - 2\]we know that the slope (m) is \[\frac{1}{4}\]and the y-intercept (b) is \[-2\].

plotting points

Plotting points on a graph is a fundamental skill for understanding and visualizing linear equations. Here’s how you do it:

• Identify the coordinates: Each point on a graph has coordinates (x, y)or in our case (u, v).
• Locate the y-intercept: This is the point where the line crosses the vertical axis. For our equation \[v = \frac{u}{4} - 2\], the y-intercept is \[-2\].So locate v = -2 on the vertical axis.
• Use the slope to find another point: The slope tells you how steep the line is. If the slope is \[\frac{1}{4}\], it means that for every 4 units you move to the right along the u axis, you move up 1 unit on the v axis. Starting from (0, -2), move up 1 unit to -1, and then 4 units to the right to 4 on the u axis. You get another point (4, -1).
• Draw the line: Once you have two points, you can draw a straight line through them. This is the graph of your linear equation.

By repeating these steps for other equations, you can graph any linear equation easily.

linear equations

A linear equation models a straight line on a graph. The general form of a linear equation with two variables is \[Ax + By = C\]where

• , , and are constants
• x is the independent variable
• y is the dependent variable

Linear equations have some distinctive features:

• They form straight lines when graphed.
• They have a constant rate of change, known as the slope.
• They can be rearranged into slope-intercept form \[y = mx + b\]for easier graphing.

For our exercise, we started with a linear equation in standard form: u - 4v = 8. We rearranged it to get the slope-intercept form v = \frac{u}{4} - 2, making it easier to graph. By understanding these basic principles, you can solve and graph any linear equation effectively.