Question

The graphs of the two equations appear to be parallel. Are they? Justify your answer by using elimination to solve the system. $$\left\\{\begin{array}{l}200 y-x=200 \\ 199 y-x=-198\end{array}\right.$$

Short answer

No, the lines are not parallel as they have different slopes and they intersect at the point (79600,398).

Step-by-step solution

Simplify the Equations

First, transform the equations to the form of y = mx + b (slope-intercept form). For the first equation, it would be \(y = \frac{1}{200}x + 1\). For the second equation, it would be \(y = \frac{1}{199}x -1.\)

Identify the Slopes

From these equations, the slope (m) can be identified. For the first equation, m = \(\frac{1}{200}\), and for the second equation, m = \(\frac{1}{199}\). These slopes are not equal, meaning these are not parallel lines.

Verify by Elimination Method

The method of elimination involves adding or subtracting the equations in order to eliminate one of the variables, which can further verify the result. In this situation, from the original system of equations subtract the second equation from the first to get \[y = 398.\] Solve for x using this new value of y in either of the given equations. Solve for x in first equation: \[ x = 200y - 200 = 79600.\] The solution for the system is (79600, 398). As the system has a unique solution, the lines are not parallel.

Key concepts

Elimination Method

When faced with a system of linear equations, the elimination method is a straightforward tool for finding the point at which the two lines intersect. The key idea behind the elimination method is to combine the equations in such a way that one of the variables is effectively 'eliminated'. This allows the solver to find the value of the remaining variable with ease.

As seen in the problem, we start by organizing both equations into the same structure. Then, we manipulate one or both equations (by either adding or subtracting) to remove one of the variables. In our example, we subtracted the second equation from the first, which removed the variable 'x' and left us with a single equation in terms of 'y'. Once we found the value of 'y', we substituted this value back into one of the original equations to find 'x'. This particular system yielded a unique solution, indicating that the lines intersect at exactly one point and are thus not parallel.

Parallel Lines

In geometry, parallel lines are lines in a plane that do not meet; that is, two lines in a plane that do not intersect or touch each other at any point. From an algebraic perspective, for the graphs of two linear equations to represent parallel lines, they must have the same slope but different y-intercepts - this is because the slope determines the angle at which the line inclines from the horizontal axis, and if two lines have the same incline but start from different points on the y-axis, they will never cross. The problem in question challenges us to determine whether two lines are parallel by examining their equations. After transforming the equations into slope-intercept form and verifying that the slopes were close but not identical, we could conclude that the lines are not parallel. Had the slopes been equal and the y-intercepts different, the lines would indeed be parallel.

Slope-Intercept Form

The slope-intercept form is incredibly beneficial for quickly understanding the key characteristics of a line, such as its slope and where it crosses the y-axis. Written as \(y = mx + b\), 'm' represents the slope while 'b' signifies the y-intercept. To solve the system of equations presented in the problem, we first converted the equations into the slope-intercept form. This not only prepared the equations for comparison to check for parallelism but also set up the scenario for applying the elimination method effectively by making the coefficients of 'x' and 'y' clearly visible. Despite the slopes of the lines in the problem being very close, the conversion to slope-intercept form revealed that they were not the same, prompting further investigation through elimination that led to the determination that the lines are not parallel because the system does have a solution.