Problem 33 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y+7=x-5$$

Step-by-Step Solution

Answer

The graph is a line with y-intercept (0, -12) and x-intercept (12, 0).

Step 1 - Rewrite the Equation in Slope-Intercept Form

First, solve for y in terms of x. Starting with the given equation:y + 7 = x - 5Subtract 7 from both sides to isolate y:y = x - 5 - 7So, the slope-intercept form is:y = x - 12

Step 2 - Identify the Slope and Y-Intercept

From the equation y = x - 12, we can identify the slope (m) and the y-intercept (b):The slope m is 1 and the y-intercept b is -12.The point (0, -12) is the y-intercept.

Step 3 - Find the X-Intercept

To find the x-intercept, set y to 0 in the equation and solve for x:0 = x - 12x = 12The point (12, 0) is the x-intercept.

Step 4 - Plot the Intercepts

Plot the points (0, -12) and (12, 0) on the coordinate plane. These are the intercepts of the graph.

Step 5 - Sketch the Line

Draw a straight line through the points (0, -12) and (12, 0) to represent the equation y = x - 12.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is a way of writing the equation so that it shows the slope and the y-intercept directly. The general form is: \[y = mx + b\] where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept, the point where the line crosses the y-axis

To rewrite an equation into slope-intercept form, solve for y. For the given exercise, \(y + 7 = x - 5\), we subtract 7 from both sides:
\[y = x - 5 - 7\] which simplifies to:
\[y = x - 12\]
Now, the equation is\[y = mx + b\]. Here, \(m = 1\) and \(b = -12\). This tells us the line has a slope of 1 and intersects the y-axis at -12. Understanding the slope-intercept form helps us quickly identify these properties and make graphing easier.

intercepts

Intercepts are the points where the line crosses the x-axis and y-axis. They are essential for drawing the line accurately.
The y-intercept is found by setting x to 0 in the equation:

• In our slope-intercept form \(y = x - 12\), substitute \(x = 0\).
  \[y = 0 - 12 = -12\]
• The y-intercept is (0, -12).

The x-intercept is found by setting y to 0:

• Set \(y = 0\) in the equation \(0 = x - 12\).
  \[0 = x - 12 \Rightarrow x = 12\]
• The x-intercept is (12, 0).

By finding these intercepts, we now have two key points to plot the line.

coordinate plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves. It has two perpendicular lines called axes:

• The horizontal axis (x-axis)
• The vertical axis (y-axis)

The point where the axes intersect is called the origin (0,0). To graph the equation \(y = x - 12\):

• First, plot the y-intercept at (0, -12)
• Next, plot the x-intercept at (12, 0)

These points help in accurately drawing the line.
Draw a straight line through (0, -12) and (12, 0) to represent the equation. This visual representation on the coordinate plane makes it easier to understand how changes in the equation affect the graph.