Question

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ -4 x+5 y=40 $$

Short answer

x-intercept: (-10, 0). y-intercept: (0, 8).

Step-by-step solution

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and then solve for x. Substituting y with 0 in $$ -4x + 5y = 40 $$ leads to $$ -4x + 5(0) = 40 $$ $$ -4x = 40 $$ Dividing both sides by -4, $$ x = -10 $$ Therefore, the x-intercept is (-10, 0).

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and then solve for y. Substituting x with 0 in $$ -4x + 5y = 40 $$ leads to $$ -4(0) + 5y = 40 $$ $$ 5y = 40 $$ Dividing both sides by 5, concludes to $$ y = 8 $$ Therefore, the y-intercept is (0, 8).

- Graph the equation

Using the intercepts found in the previous steps, plot the points (-10, 0) and (0, 8) on a coordinate plane. Then, draw a straight line through these two points to represent the equation $$ -4x + 5y = 40 $$ The line should extend through the plotted points and continue to both directions infinitely.

Key concepts

Intercepts

Intercepts are key points where a graph crosses the axes. These points can be very useful for graphing linear equations. We have two types of intercepts: x-intercept and y-intercept.
Finding the x-intercept involves setting y to 0 and solving for x. In this case, substituting y with 0 in \[ -4x + 5(0) = 40 \] leads to \[ -4x = 40 \]. By solving \[ x = -10 \], we find the x-intercept at point (-10, 0).
Similarly, to find the y-intercept, set x to 0 and solve for y. Substituting x with 0 in \[ -4(0) + 5y = 40 \] leads to \[ 5y = 40 \]. Dividing both sides by 5, we get \[ y = 8 \]. Thus, the y-intercept is at point (0, 8).
Remember the intercepts' coordinates are given as (x, 0) for the x-intercept and (0, y) for the y-intercept. These points are crucial for sketching the graph correctly.

Graphing Equations

Graphing equations start with plotting the intercepts. Starting with the x-intercept and y-intercept simplifies the process.
Once we have the intercepts, use these points to plot them on the coordinate plane. For our example, plot the points (-10, 0) and (0, 8).
After plotting, draw a straight line through these points. This line represents your equation \[ -4x + 5y = 40 \].
It's essential to make sure this line extends through the plotted points and continues infinitely in both directions. This step visualizes the linear relationship represented by the equation.
The straight line makes it easier to see where the equation intersects with the axes, providing a clear visual of the x-intercept and y-intercept.

Coordinate Plane

The coordinate plane is a crucial tool for graphing linear equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
The point where these axes intersect is called the origin, with coordinates (0, 0). The coordinate plane is divided into four quadrants, each representing a combination of positive and negative values for x and y.
Learning to navigate this plane effectively is key.

• The first quadrant has positive x and y values.
• The second quadrant has negative x and positive y values.
• The third quadrant has negative x and y values.
• The fourth quadrant has positive x and negative y values.

When plotting points, start at the origin. Move horizontally to the x-coordinate, then vertically to the y-coordinate. Plotting our intercepts (-10, 0) and (0, 8) correctly on the coordinate plane ensures accuracy in drawing our line.
This precision ensures we correctly represent the equation on the graph.