Problem 27 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(y = 9/5x\).

Step 2: 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.

Step 3: 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 \(y = 9/5x + 20\).

Step 4: 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.

Key Concepts

Rearranging Equations

Rearranging equations is a foundational step in solving systems of linear equations. It means transforming the original equations into a more convenient form, typically like the slope-intercept form: \(y = mx + c\).
This simplifies the process of comparing and solving the equations. In our original system:

• First equation: \(100 - 9x = 5y\)
• Second equation: \(0 = 5y - 9x\)

By rearranging the first equation, we add \(9x\) to each side and divide by 5:

• \(5y = 9x + 100\)
• \(y = \frac{9}{5}x + 20\)

Similarly, for the second equation, after corrections, it rearranges to the same line:

• \(0 = 5y - 9x\)
• \(y = \frac{9}{5}x + 20\)

By converting equations into this form, we can directly compare slopes and y-intercepts. The rearranging step often uncovers misunderstandings or errors in set-up, which is crucial for finding accurate solutions.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. In the context of linear equations, parallelism occurs when two lines have the same slope but different y-intercepts. This means that they rise and fall at the same rate but at different starting points, ensuring they never meet.
From our rearranged equations:

• \(y = \frac{9}{5}x + 20\)
• Initially thought: \(y = \frac{9}{5}x\)

These were initially considered parallel because they shared the slope \(m = \frac{9}{5}\) but seemed to have differing intercepts.
However, verifying and correcting mistakes showed they were not just parallel but identical. Always check both slope and intercept precision.

Infinite Solutions

When dealing with linear systems, infinite solutions denote a special case. This occurs when two equations represent the exact same line. Therefore, every point on the line satisfies both equations.
Correctly identifying infinite solutions requires showing:

• The same slope \(m\)
• And the same y-intercept \(c\)

As seen, both equations settled to \(y = \frac{9}{5}x + 20\). They coincide completely, thus demonstrating that there is not just one point of intersection, but infinitely many.
Such results suggest both representations stem from the same linear relationship and were possibly approximations of each other from the start.
Rechecking original identities ensures no oversight, securing the solution's integrity.