Problem 68 - Precalculus with Limits

Question

Think About It In Exercises 67 and \(68,\) the graphs of the two equations appear to be parallel. Yet, when you solve the system algebraically, you find that the system does have a solution. Find the solution and explain why it does not appear on the portion of the graph shown. $$ \left\\{\begin{aligned} 21 x-20 y &=0 \\ 13 x-12 y &=120 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \(x = 666.67\) and \(y = 700\). The intersection point does not show up on the graph because the x and y resolution might be too small to show far points unless the means of plotting the graph is set to portray these points.

Step 1: Simplify the equations

We can simplify the equations by moving terms and having a form of \(y = mx + b\). The simplified form for the first equation would be \(y = 1.05x\). Likewise, the simplified form for the second equation would be \(y = 1.08333x - 10\).

Step 2: Solve the system algebraically

To find the intersection point of the two lines, we solve the system of equations by setting the two equations equal: \(1.05x = 1.08333x - 10\). Solving this, we get \(x = 666.67\).

Step 3: Find the corresponding y-value

We can find the value of y by substituting \(x = 666.67\) into the equation \(y = 1.05x\). This gives \(y = 700\). So, the solution to the system is \(x = 666.67\) and \(y = 700\).

Step 4: Explain the reason

The reason the intersection point does not appear to be on the portion of the graph shown is likely due to the x and y resolution used when plotting the graphs. With a lower resolution, the intersection point could be missed unless the software or means of plotting the graph is set to accurately illustrate points far from the origin. In this case, because the x-value of the intersection point is quite high (\(x = 666.67\)), it would not appear on a graph with a smaller range.

Key Concepts

Parallel Lines

When studying algebra and specifically, systems of equations, we often encounter the concept of parallel lines. In a graphical sense, parallel lines are lines in a plane that never meet; they have the same slope but different y-intercepts. In algebraic terms, this means that the equations representing the lines have identical coefficients in front of the variable x, reflecting identical slopes, but different constant terms, reflecting different y-intercepts.

However, sometimes two lines may appear to be parallel on a graph due to limitations in viewing or plotting capabilities, when in fact, they may intersect at a point that is not immediately visible. This is particularly common when the point of intersection lies far from the origin or outside the typical viewing window of a graphing utility. A thorough algebraic solution can reveal the exact intersection point, even when graphical methods fail to show it.

Algebraic Solution

Solving a system of equations algebraically is a reliable method to find out whether two lines intersect and if so, to determine the exact point of their intersection. An algebraic solution involves manipulation of the equations until we can isolate the variables and solve for their values. The process starts with simplifying each equation to a form where one variable is expressed in terms of the other (like the slope-intercept form, y = mx + b).

Once in this form, we can set the equations equal to each other since at the point of intersection, both y values from each equation will be the same. By solving for x, we find the precise point on the x-axis where the lines meet. Substituting this x-value into either equation then gives us the corresponding y-value, thus providing the solution to the system, which represents the intersection point (if it exists).

Graphical Representation

A graphical representation of a system of equations is a very visual approach to understanding the relationship between two algebraic expressions. By plotting each equation on a graph, we can visually assess whether and where the lines intersect.

However, the graphical method has limitations, such as resolution and scale. The intersection point of two lines may not be visible if it lies outside the plotted range. This is an essential aspect of using graphing technology; we must make sure that the scale is set appropriately to capture relevant points of intersection, especially for large x or y values, as was the case in the exercise. In educational settings, especially when teaching the concept of systems of equations and their solutions, it is crucial to remind students to consider these limitations and always verify their graphical findings with an algebraic approach.