Problem 25 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$x-y=8$$

Step-by-Step Solution

Answer

The graph intersects the y-axis at (0, -8) and the x-axis at (8, 0).

Step 1: Rearrange the Equation

Rewrite the given equation in the slope-intercept form, which is y = mx + b. By adding y to both sides of the equation, we get:x = y + 8Then, subtract 8 from both sides to solve for y:y = x - 8

Step 2: Identify the Slope and Y-intercept

From the equation y = x - 8, identify the slope and the y-intercept. Here, the slope (m) is 1, and the y-intercept (b) is -8. This means the line crosses the y-axis at (0, -8).

Step 3: Find the X-intercept

To find the x-intercept, set y to 0 in the equation y = x - 8:0 = x - 8Solve for x:x = 8This means the line crosses the x-axis at (8, 0).

Step 4: Sketch the Graph

Draw the coordinate axes. Plot the y-intercept at (0, -8) and the x-intercept at (8, 0). Connect these two points with a straight line, extending the line in both directions.

Step 5: Label the Intercepts

Clearly label the points where the line intersects the axes: (0, -8) on the y-axis and (8, 0) on the x-axis.

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to represent a straight line. The equation is written as: y = mx + b.

Here,

• 'y' is the dependent variable
• 'x' is the independent variable
• 'm' represents the slope of the line
• 'b' is the y-intercept, the point where the line crosses the y-axis

To rewrite any linear equation in slope-intercept form, you need to isolate 'y' on one side of the equation. For the given equation x - y = 8, adding 'y' to both sides results in x = y + 8. Then, subtract 8 from both sides to get y = x - 8. This is now in slope-intercept form, where the slope (m) is 1 and the y-intercept (b) is -8.

x-intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the value of 'y' is 0.

To find the x-intercept:

• Set y to 0 in the equation
• Solve for x

In our equation y = x - 8, setting y to 0 gives 0 = x - 8. Adding 8 to both sides, we find x = 8. So, the x-intercept is at (8, 0).
This means that the line crosses the x-axis at this point.

y-intercept

The y-intercept is the point where the line crosses the y-axis. At this point, the value of 'x' is 0.

To find the y-intercept:

• Set x to 0 in the equation
• Solve for y

In our equation y = x - 8, setting x to 0 gives y = 0 - 8. Simplifying, we find y = -8. So, the y-intercept is at (0, -8).
This means that the line crosses the y-axis at this point.