Question

The graphs of the two equations appear to be parallel. Are they? Justify your answer by using elimination to solve the system. $$\left\\{\begin{array}{l}25 x-24 y=0 \\ 13 x-12 y=120\end{array}\right.$$

Short answer

Yes, the graphs of the equations \(25x - 24y = 0\) and \(13x - 12y = 120\) are parallel. This is shown by their having the same slope but different y-intercepts, and the lack of solutions when the system of equations is solved using elimination.

Step-by-step solution

Writing down the system of equations

The given system of equations is: \[\begin{align*} 25x - 24y &= 0, \ 13x - 12y &= 120. \end{align*}\]

Simplifying the equations to identify the coefficients

First, simplify the equations so that the coefficients of \(x\) and \(y\) are easier to compare. The simplified forms of the original equations are: \[\begin{align*} x - \(\frac{24}{25}\)y &= 0, \ x - \(\frac{12}{13}\)y &= \(\frac{120}{13}\). \end{align*}\]

Perform the elimination

Subtract the second equation from the first: \(\[\] (x - \(\frac{24}{25}\)y) - (x - \(\frac{12}{13}\)y) = 0 - \(\frac{120}{13}\)\). This simplifies to \(0 = \(\frac{120}{13}\) - 0\), which is a contradiction. Therefore, this system of equations has no solution.

Conclusion

The lines represented by the two equations \(25x - 24y = 0\) and \(13x - 12y = 120\) are parallel because they have the same slope but different y-intercepts, and the system of equations has no solutions when using the method of elimination. However, their slopes are not in exactly the same ratio as the original coefficients, due to the constant terms in the equations.

Key concepts

Elimination Method

The elimination method is a systematic procedure used to solve a system of linear equations. This method involves adding or subtracting the equations in order to eliminate one of the variables, making it possible to solve for the other variable. For clarity, consider the following steps:

• First, align the equations so that like terms are vertically stacked.
• Then, if necessary, multiply one or both of the equations by a constant to obtain coefficients that will cancel out a variable when the equations are added or subtracted.
• Next, add or subtract the equations to eliminate one variable.
• Solve for the remaining variable.
• Substitute the value of the found variable into one of the original equations to find the value of the other variable.

Sometimes, through elimination, we may reach an equation that makes no sense, such as '0 = a non-zero number'. This indicates that the system has no solution, which typically means the lines are parallel.

Parallel Lines

In coordinate geometry, parallel lines have the same slope, meaning they never intersect. When we are given two linear equations and asked to determine whether their graphs are parallel, we compare their slopes. If the slopes are identical, and the y-intercepts are different, the lines are parallel.

To find the slope of a line given by an equation in the form 'ax - by = c', we can rearrange it into slope-intercept form 'y = mx + b', where 'm' is the slope. If two equations have the same value for 'm' but different 'b' (y-intercept), their graphs will be parallel lines on the Cartesian plane. This is important because, in a system of equations, parallel lines mean that there is no single point of intersection, and therefore, no solution that satisfies both equations.

Algebraic Contradictions

Encountering an algebraic contradiction, such as '0 = 5', signifies that the algebraic process has led to an impossible situation. This usually occurs when solving a system of linear equations where the lines represented by these equations do not intersect -- namely, they are parallel. An algebraic contradiction indicates that there is no solution to the system (also known as an inconsistent system).

In the context of solving equations by elimination, reaching a contradiction, like '0 = a non-zero constant', unequivocally proves that there are no points that satisfy both equations. Consequently, it doesn't matter which values we assign to the variables; the equations will never equal the same thing at the same time. This is an essential concept in algebra because it helps to distinguish between systems that are solvable and those that are not.

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The solution to such a system is the set of values for the variables that satisfies all equations simultaneously. Solutions can be represented as a point on a graph, where the lines or planes of the equations meet.

When dealing with a system of linear equations, there are three possible outcomes:

• A unique solution, where the lines intersect at one point.
• No solution, where the lines are parallel and never intersect.
• Infinitely many solutions, where the lines are coincident, essentially laying on top of each other.

Understanding how to analyze and solve these systems is fundamental in algebra. Methods for solving them include graphing, substitution, elimination, and matrix operations.