Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Use intercepts and a checkpoint to graph each equation. $$3 x-2 y=-7$$

Step-by-Step Solution

Answer

The x-intercept of the equation is \(-7/3, 0\), the y-intercept is \(0, 7/2\) and a possible checkpoint is \((0, -7/2)\). These points help drawing the graph of the equation \(3x - 2y = -7\).

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. This gives the equation \(3x - 2(0) = -7\) which simplifies to \(3x = -7\), giving \(x = -7/3\). So the x-intercept is \(-7/3, 0\).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Now we have the equation \(3(0) - 2y = -7\) which simplifies to \(-2y = -7\), giving \(y = 7/2\). So the y-intercept is \(0, 7/2\).

Step 3: Plot the intercepts

Now that we have both intercepts, plot these on the coordinate plane. The x-intercept is at \(-7/3, 0\) and the y-intercept is at \(0, 7/2\).

Step 4: Find a checkpoint

Now we will select a checkpoint not on the x or y axes. A simple checkpoint to calculate would be (1,1). Substituting these values into the equation gives \(3(1) - 2(1) = -7\), which simplifies to \(3 - 2 = 1\). Yet, -7 does not equal to 1. This indicates that the checkpoint (1,1) does not lie on the line. So, a correct checkpoint could be (0,1), substituting it into the equation results in 3(0) - 2(1) = -7 which simplifies to -2 = -7. Hence, the checkpoint could be (0, -7/2).

Step 5: Draw the Line

With the intercepts and a checkpoint plotted, draw a line through these points. The line through the points will represent the graph of equation \(3x - 2y = -7\).

Key Concepts

X-Intercept

Understanding the x-intercept is crucial while graphing linear equations. It's the point where the graph crosses the x-axis. To find it, you simply set the y-value to 0 and solve the equation for x. For example, in the equation \(3x - 2y = -7\), when y is 0, solving \(3x = -7\) gives us the x-intercept as \(-7/3, 0\). It indicates where the line touches the x-axis, which is essential for plotting the graph.

To grasp the concept, imagine you're walking along a straight path that represents a linear equation. The x-intercept is where your path starts or ends at the horizontal axis if you were moving purely left or right.

Y-Intercept

In contrast to the x-intercept, the y-intercept is where the equation's graph intersects the y-axis. To find the y-intercept, set the x-value to zero and solve for y. Taking our equation \(3x - 2y = -7\), with x set to zero, we find \(-2y = -7\), which simplifies to \(y = 7/2\). Hence, the y-intercept is \(0, 7/2\).

Think of the y-intercept as the height at which you would begin or finish your journey if you were traveling vertically along your path. It's an invaluable starting point for sketching the line representing the equation on the graph.

Coordinate Plane

The coordinate plane is the 'stage' on which we plot equations. It consists of a horizontal line (x-axis) and a vertical line (y-axis) that intersect at a right angle at the origin point (0,0). Every point on the plane is determined by an x (horizontal) and y (vertical) coordinate.

To visualize, think of the coordinate plane as a map with a starting point at the center. The x-axis tells you how far east or west you travel, and the y-axis how far north or south. By plotting the x and y-intercepts on this 'map,' you're setting the stage to draw the full path of the linear equation.

Linear Equation Checkpoints

Checkpoints in graphing are like samples to check the accuracy of the graph. After plotting the x and y-intercept, we choose additional points to ensure our line is correct. These should not be on the x or y-axis. For the equation \(3x - 2y = -7\), the point (1,1) is not suitable as it doesn't satisfy the equation. Instead, (0,-3.5) is a correct checkpoint as it fulfills the equation.

Imagine checkpoints as confirmations on your path – if you can pass through them as you draw your line on the graphing 'map', you're on the right track. They serve as extra evidence that the line is plotted correctly.