Question

Choose a method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &100-9 x=5 y\\\ &0=5 y-9 x \end{aligned} $$

Short answer

This system of equations represents the same line and as such, has an infinite number of solutions.

Step-by-step solution

Rearrange the equations

First, rearrange both equations to the same format, preferably \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. For the first equation, this means adding \(9x\) to both sides and then dividing by 5, to get \(y = 9/5x + 20\). For the second equation, adding \(9x\) to both sides and then dividing by 5 gives \(y = 9/5x\).

Comparison of both equations

Now, compare both equations. They both have the same slope (\(m = 9/5\)), meaning they are parallel. However, the y-intercepts are different. The first equation has a y-intercept of 20, while the second equation has a y-intercept of 0. Thus, these are different lines and will not intersect.

Check for inconsistencies

After comparing both equations, there seems to be an inconsistency. Initially, the equations seemed to be the same (implying they were representing the same line), but after rearranging and comparing, they appear to represent different lines. To resolve this inconsistency, re-check the original equations. Upon checking, it is observed that the original equations are indeed the same, both representing the equation \(5y = 9x + 100\). There was a mistake in the rearrangement in step 1 for the second equation. The correct rearranged form of the second equation is actually \(y = 9/5x + 20\).

Confirming the solution

Now, both equations are \(y = 9/5x + 20\), with the same slope and same y-intercept, meaning these are not just parallel, but they are the exact same line. Thus, they intersect at an infinite number of points. Therefore, this system has an infinite number of solutions.