Problem 65 - Elementary Algebra Within Reach

Question

In Exercises \(63-66\), sketch the graph of the equation. $$ 0.3 x-0.2 y=0.8 $$

Step-by-Step Solution

Answer

The x-intercept is \( \frac{0.8}{0.3}\) and the y-intercept is \( \frac{-0.8}{0.2}\). Connect these two points on the graph to plot the line.

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 and solve for x in the equation. In this case, when y=0, \(0.3x-0.2(0)=0.8\), then we solve for x so \(x = \frac{0.8}{0.3}\).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 and solve for y in the equation. In this case, when x=0, \(0.3(0)-0.2y = 0.8\), then we solve for y so \(y = \frac{-0.8}{0.2}\).

Step 3: Sketch the graph

Plot the x-intercept and y-intercept we found in the previous steps on the Cartesian plane. Then connect these two points with a straight line. This line is the graph of the equation.

Key Concepts

X-Intercept

The x-intercept of a graph is the point at which the line crosses the x-axis of the Cartesian plane. This is the key to beginning our sketch. To find it, we set the y-variable to 0, because on the x-axis, the value of y is always 0. In our exercise, the equation given is 0.3x - 0.2y = 0.8.

Following the step-by-step solution, we put y to 0 and solve for x, giving us the equation 0.3x = 0.8. To get x on its own, divide both sides by 0.3, resulting in \( x = \frac{0.8}{0.3} \). Simplifying this gives us the precise location of the x-intercept, which is an integral part of graphing the linear equation.

Y-Intercept

Conversely, the y-intercept is where the line meets the y-axis. To identify this point, we keep the x-variable at 0, as any point on the y-axis has an x-value of 0. Referring back to our original equation for our exercise, we set x to 0, yielding 0.2y = -0.8. We then isolate y by dividing both sides by -0.2 to obtain \( y = \frac{-0.8}{0.2} \).

This calculation provides us the y-intercept, another essential component for sketching our graph. It's important to note that the y-intercept reflects the starting value of y when x is 0; it indicates the height at which the line crosses the y-axis.

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, consists of two perpendicular axes, usually labeled x (horizontal) and y (vertical). Every point on this plane can be described using an ordered pair of numbers, referred to as coordinates (x, y).

The intersection of the axes represents the origin, where both x and y are zero. The plane is divided into four quadrants, each with unique sign combinations for their coordinates. The Cartesian plane is the stage on which we'll draw graphs, like the linear equation from our exercise. Understanding this plane is vital to correctly plotting intercepts and other points.

Sketching Graphs

When it comes to sketching graphs of linear equations, you start with plotting the x and y intercepts on the Cartesian plane. These two points are usually enough because a line is the shortest path between two points and the graph of a linear equation is always a line.

After marking the intercepts, we draw a straight line through them, which extends in both directions, unless domain or range is limited. The line's slope, which can be calculated if additional points are known, indicates the steepness and the direction the line takes. By correctly plotting and connecting these essential points, we create an accurate representation of the equation's graph.