Problem 46 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=-35 x-1498$$

Step-by-Step Solution

Answer

The y-intercept is (0, -1498) and the x-intercept is (-42.8, 0). Sketch the line through these points.

Step 1: Identify the Equation Form

The given equation is in slope-intercept form, which is written as y=mx+c In this equation, m=-35 c=-1498

Step 2: Find the y-intercept

The y-intercept occurs when x = 0. Substitute x = 0 into the equation: y = -35(0) - 1498 y = -1498 So, the y-intercept is (0, -1498).

Step 3: Find the x-intercept

The x-intercept occurs when y = 0. Set y = 0 and solve for x: 0 = -35x - 1498 35x = -1498 x = -1498/35 x = -42.8 So, the x-intercept is (-42.8, 0).

Step 4: Sketch the Graph

Plot the y-intercept (0, -1498) and the x-intercept (-42.8, 0) on the graph. Draw a straight line through these points since the equation represents a straight line.

Step 5: Label the Intercepts

Clearly label the intercepts on the graph: the y-intercept at (0, -1498) and the x-intercept at (-42.8, 0).

Key Concepts

y-intercept

The y-intercept is the point where the graph of a linear equation crosses the y-axis. This happens when the value of x is zero. To find the y-intercept, you substitute x = 0 into the equation and solve for y. In the given equation, \( y = -35x - 1498 \), you would set x to zero and get \( y = -1498 \). So, the y-intercept is at the point (0, -1498). This point indicates where the line will touch the y-axis. Understanding the y-intercept helps in easily sketching and analyzing the graph of the equation.

x-intercept

The x-intercept is the point where the graph of a linear equation crosses the x-axis. This happens when the value of y is zero. To find the x-intercept, you set y = 0 in the equation and solve for x. For the equation \( y = -35x - 1498 \), you set y to zero, giving you \( 0 = -35x - 1498 \). Solving for x, you get \( x = -42.8 \). So, the x-intercept is at the point (-42.8, 0). Knowing the x-intercept helps in plotting the line accurately and understanding its behavior on the graph.

slope-intercept form

The slope-intercept form of a linear equation is written as \( y = mx + c \), where m represents the slope of the line and c represents the y-intercept. This form is very useful because it immediately gives you the slope and the y-intercept. The slope, m, shows how steep the line is, and whether it tilts upward or downward. In the equation \( y = -35x - 1498 \), the slope m is -35, showing that the line slopes downward. The y-intercept c is -1498, indicating where the line crosses the y-axis. Understanding the slope-intercept form helps in quickly sketching and interpreting the graph of the line.