Problem 72 - Intermediate Algebra

Question

Two airplanes, flying at the same altitude, are tracked using the same coordinate system on a radar screen. One plane is following a path described by the equation \(y=0.4 x-2,\) and the other is following a path described by the equation \(2 x=5 y+7 .\) Graph both equations on the same coordinate system. Is there a possibility of a collision?

Step-by-Step Solution

Answer

No collision; the paths are parallel.

Step 1: Converting the Second Equation to Slope-Intercept Form

The equation given for the second airplane is not in slope-intercept form. Start by solving \(2x = 5y + 7\) for \(y\). Subtract 7 from both sides to get \(2x - 7 = 5y\). Now divide every term by 5: \(y = \frac{2}{5}x - \frac{7}{5}\). The slope-intercept form of the second equation is \(y = \frac{2}{5}x - \frac{7}{5}\).

Step 2: Graphing the First Equation: y = 0.4x - 2

To graph the first equation \(y = 0.4x - 2\), start by identifying the slope \(m = 0.4\) and the y-intercept \(b = -2\). Plot the point (0, -2) on the y-axis. Use the slope \(0.4\) (which is \(\frac{2}{5}\)) to find another point by rising 2 units and running 5 units from the intercept, plotting this second point.

Step 3: Graphing the Second Equation: y = \frac{2}{5}x - \frac{7}{5}

Identify the slope \(m = \frac{2}{5}\) and the y-intercept \(b = -\frac{7}{5}\) from the equation \(y = \frac{2}{5}x - \frac{7}{5}\). Plot the y-intercept point \(\left(0, -\frac{7}{5}\right)\). From this point, use the slope to rise 2 units and run 5 units to identify another point, then plot that point as well.

Step 4: Checking for Intersection

To check if the planes might collide, find the intersection of the two lines. Set \(0.4x - 2 = \frac{2}{5}x - \frac{7}{5}\). Convert 0.4 into a fraction \(\frac{2}{5}x\), so the equation becomes \(\frac{2}{5}x - 2 = \frac{2}{5}x - \frac{7}{5}\). Since both sides are equal after eliminating \(\frac{2}{5}x\), there is no specific solution, as the equations represent parallel lines with different y-intercepts.

Key Concepts

Graphing Lines

Graphing lines on a coordinate plane is a fundamental skill in algebra and helps visualize relationships between variables. When graphing a line from an equation, we're essentially plotting all the points that satisfy that equation. To graph a line given in slope-intercept form, the process is straightforward:

• Start by identifying the y-intercept, which is the point where the line crosses the y-axis. This is where you plot your first point.
• Then, use the slope to find another point. The slope is a ratio that describes how steep the line is. It tells us how much "rise" (vertical change) we should have for a given "run" (horizontal change).
• Connect these points to draw the line, ensuring your ruler or straight edge keeps it straight.

Graphing is a visual way to understand how different equations relate by showing the slope and y-intercept of the line directly on the graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is an efficient way to describe a line. An equation in this form looks like: \[y = mx + b\], where:

• \(m\) represents the slope of the line. This tells us the direction and steepness of the line. A positive slope means the line ascends, while a negative slope means it descends.
• \(b\) is the y-intercept. This is the value of \(y\) when \(x = 0\), indicating where the line crosses the y-axis.

For example, in the equation \(y = 0.4x - 2\), the slope is 0.4 and the y-intercept is -2. This means for every 5 units you move right horizontally, you move 2 units up vertically. Writing equations this way makes it simple to graph and interpret them.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. One key characteristic of parallel lines is that they have the same slope but different y-intercepts, meaning they run overall in the same direction while staying a fixed distance apart.
To determine if two lines are parallel, examine their equations:

• Compare their slopes. If the slopes are identical and y-intercepts different, the lines are parallel.

In the example, the two airplanes' paths are modeled by the equations \(y = 0.4x - 2\) and \(y = \frac{2}{5}x - \frac{7}{5}\). Both slopes are \(\frac{2}{5}\), indicating they are parallel. Thus, they travel in the same direction forever without meeting.