Problem 24 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$x-y=4$$

Step-by-Step Solution

Answer

The intercepts are at (4, 0) and (0, -4).

Step 1 - Convert to Slope-Intercept Form

Start by converting the given equation into slope-intercept form where it looks like \(y = mx + b\). Given the equation \(x - y = 4\), we need to solve for \(y\):\[ x - y = 4 \implies -y = -x + 4 \implies y = x - 4 \].

Step 2 - Identify the Slope and Intercept

From the equation \(y = x - 4\), identify the slope \(m\) and the y-intercept \(b\). Here, the slope \(m = 1\) and the y-intercept \(b = -4\). This tells us that the line crosses the y-axis at \( (0, -4) \).

Step 3 - Find the X-Intercept

The x-intercept occurs when \(y = 0\). Set \(y = 0\) in the equation \(x - y = 4\):\[ x - 0 = 4 \implies x = 4 \].Thus, the x-intercept is at \( (4, 0) \).

Step 4 - Plot the Intercepts and Draw the Line

Plot the y-intercept at \( (0, -4) \) and the x-intercept at \( (4, 0) \) on a graph. Join these two points with a straight line, extending it in both directions.

Step 5 - Label the Intercepts

Label the points where the line crosses the axes on the graph. The x-intercept at \( (4, 0) \) and the y-intercept at \( (0, -4) \).

Key Concepts

slope-intercept form

The slope-intercept form is a way of writing the equation of a straight line. This form is helpful because it clearly shows the slope of the line and the y-intercept, making it easier to graph the equation.

The general formula for the slope-intercept form is:

\[y = mx + b\]In this formula:

• \(m\): Represents the slope of the line. The slope indicates how steep the line is, showing the rate at which y changes with respect to x.


• \(b\): Represents the y-intercept of the line. This is the point where the line crosses the y-axis. It's the value of y when x is zero.

For the given equation, we start with the standard form: \(x - y = 4\)
By isolating y, we convert it into slope-intercept form:\[x - y = 4 \ \implies -y = - x + 4 \ \implies y = x - 4\]Here, it's clear that:

• The slope \(m\) is 1


• The y-intercept \(b\) is -4

graphing equations

Graphing equations involves plotting points on a coordinate plane and drawing a line through these points to represent the equation. We'll use the slope and intercepts to graph the line.

To graph \(y = x - 4\):

First, plot the y-intercept:

• The y-intercept \((0, -4)\) is where the line crosses the y-axis.

Next, use the slope to find another point:

• The slope \(m = 1\) means that for every one unit increase in x, y increases by one unit.


• Starting from the y-intercept, go up 1 unit and right 1 unit to find another point on the line. This would be \((1, -3)\).

Finally, draw the line:

• Connect the points \((0, -4)\), \((1, -3)\), and any other points along the slope with a straight line.


• Extend the line in both directions, adding arrows at the ends to show that it continues indefinitely.

finding intercepts

Intercepts are the points where the graph crosses the axes. There are two types of intercepts: x-intercepts and y-intercepts.

The y-intercept is found where the line crosses the y-axis. This happens when x is zero:

• For \(y = x - 4\), set x to 0: \[y = 0 - 4 \implies y = -4\].


• So, the y-intercept is \((0, -4)\).

To find the x-intercept, look for the point where the line crosses the x-axis. This occurs when y is zero:

• From the original equation \(x - y = 4\), set y to 0: \[x - 0 = 4 \implies x = 4\].


• So, the x-intercept is \((4, 0)\).

These intercepts are useful for graphing the line and understanding the equation's behavior. Plot the intercepts on the graph to visualize where the line crosses the axes.

By knowing both intercepts, you can draw an accurate graph without needing additional points. This makes graphing equations quicker and simpler.