Problem 41 - Contemporary Precalculus

Question

Graph the equation. Label all intercepts. $$3 x-2 y=0$

Step-by-Step Solution

Answer

Question: Graph the equation \(3x-2y=0\) and label all intercepts. Answer: The graph of the equation \(3x-2y=0\) is a line passing through the points \((0,0)\) and \((2,3)\). Both the x-intercept and y-intercept occur at the point \((0,0)\).

Step 1: Find the x-intercept

To find the x-intercept, we set \(y=0\) and solve for \(x\): $$3x - 2(0) = 0$$ $$3x = 0$$ $$x = \frac{0}{3} = 0$$ So the x-intercept is at the point \((0,0)\).

Step 2: Find the y-intercept

To find the y-intercept, we set \(x=0\) and solve for \(y\): $$3(0) - 2y = 0$$ $$-2y = 0$$ $$y = \frac{0}{-2} = 0$$ So the y-intercept is also at the point \((0,0)\).

Step 3: Plot the intercepts and draw the line

Since both the x-intercept and y-intercept are at the point \((0,0)\), this line passes through the origin. In order to draw the line, we can find another point on the line. When \(x=2\), we can find the corresponding value of \(y\): $$3(2) - 2y = 0$$ $$6 - 2y = 0$$ $$-2y = -6$$ $$y = \frac{-6}{-2} = 3$$ So the point \((2,3)\) is also on the line. Now we can plot the points \((0,0)\) and \((2,3)\) along with the line passing through them.

Step 4: Label the intercepts

Finally, we label the intercepts on the graph. Since both the x-intercept and y-intercept occur at the point \((0,0)\), we simply label this point as the "x-intercept and y-intercept". The final graph of the equation \(3x - 2y = 0\) is a line passing through the points \((0,0)\) and \((2,3)\) with the intercepts labeled.

Key Concepts

Intercepts

When graphing a linear equation, one of the first steps is to identify the intercepts, which are the points where the graph crosses the x-axis and y-axis. Intercepts are essential because they give us a starting point to draw the line on the graph.

• X-intercept: This occurs where the line crosses the x-axis. At this point, the value of y is zero. To find the x-intercept for a given equation, set y to zero and solve for x.
• Y-intercept: Conversely, it occurs where the line crosses the y-axis. Here, the value of x is zero. To find the y-intercept, set x to zero and solve for y.

In our exercise, both intercepts [(0,0)] are the same because when we solve for both, we find that the line passes through the origin. This means that the line crosses the axes at the origin. This step is essential because intercepts provide fixed points that help sketch the entire line.

Plotting Points

Plotting points is a crucial part of graphing linear equations. Once you find the intercepts, you can plot these points on a graph. This gives you concrete visual markers to draw the line.

After plotting the intercepts, it can be beneficial to find additional points to ensure the accuracy of your line. To do this, select another value for x and compute the corresponding y, or vice versa.

This exercise found an additional point, (2, 3), by setting x to 2 and solving for y. Plotting both the intercepts and this additional point provides two sure points through which the line must pass. Observing that multiple points fit the same linear path confirms the line's direction and slope.

After plotting, connect the dots with a straight line. Make sure the line extends through all areas of the graph up to the axes to show its endless nature.

Linear Equations

Linear equations describe a straight line on a graph. They have no curves or bends and can be presented in various forms such as the slope-intercept form, point-slope form, and the standard form.

In this exercise, the equation provided is in standard form, which looks like Ax + By = C.
The example's standard form is 3x - 2y = 0. The characteristics of a straight line include a constant slope and continuous pattern throughout the graph. Linear equations are central to algebra because they represent a constant rate of change. They form the foundation for more complex topics in mathematics.

Understanding how to manipulate and graph linear equations empowers students to visualize relationships between variables. Practicing with diverse problem sets helps in grasping the slope, intercepts, and other properties of linear equations.