Problem 39 - Algebra 1

Question

a. Find a value of \(n\) so that the linear system has infinitely many solutions. b. Find a value of \(n\) so that the linear system has no solution. c. Graph both results. $$ \begin{aligned}&x-y=3\\\&4 x-4 y=n\end{aligned} $$

Step-by-Step Solution

Answer

When \(n = 12\), the system of equations has infinitely many solutions, and when \(n = 10\), the system has no solution.

Step 1: Find a value of \(n\) where the system has infinitely many solutions.

For the system of equations to have infinitely many solutions, the two equations must be multiples of each other. The second equation can be simplified to \(x - y = n/4\). Now, for the system to have infinitely many solutions, \(n/4 = 3\), which implies \(n = 12\). Therefore, when \(n = 12\), the system has infinitely many solutions.

Step 2: Find a value of \(n\) where the system has no solution

For the system of equations to have no solution, the lines represented by the equations must be parallel. This occurs when the coefficients of \(x\) and \(y\) in the two equations are the same, but the constants are different. The coefficients of \(x\) and \(y\) are already the same in the two equations. By choosing a \(n\) such that \(n/4\) is different from 3, we can ensure that the system has no solution. For example, when \(n = 10\), the system has no solution.

Step 3: Graph both results

Plot the equations \(x - y = 3\), \(x - y = n/4\) where \(n = 12\) and \(n = 10\). The graph when \(n = 12\) will show two coincident lines showing infinitely many solutions, while the graph when \(n = 10\) will show two parallel lines showing no solution.

Key Concepts

Infinitely Many Solutions

When we talk about a system of linear equations having infinitely many solutions, it means that the lines represented by these equations completely overlap on a graph. Essentially, each point on one line is also a point on the other. Imagine you have an equation that defines one line, and you simply took that equation and multiplied it by a constant. For example, if your first equation is \( x - y = 3 \), and you multiply everything by 4, you get \( 4x - 4y = 12 \), which is precisely the situation here when \( n = 12 \) in our given equation.

In this specific situation, both lines 'say' the exact same thing—they are identical! So whichever value of \( x \) or \( y \) satisfies one equation, it satisfies the other. This means there's not just one unique solution; rather, every point on that line is a solution!

Parallel Lines

Parallel lines are lines that never meet, no matter how far we extend them. This concept is crucial when discussing systems of linear equations that have no solutions. If two lines are parallel, it means they have the same slope. What's different about them is their y-intercepts.

This is why, in our problem, when \( n / 4 \) is not equal to \( 3 \), we get a pair of parallel lines. The first equation \( x - y = 3 \) and the second equation, simplified to \( x - y = n / 4 \), both have the structure that makes them parallel when their right-hand sides differ.

Thus, when you choose \( n = 10 \), it makes \( x - y = 2.5 \) (since \( n / 4 = 2.5 \)), resulting in lines that never intersect. Because they are parallel, there is no single point \((x, y)\) that satisfies both equations simultaneously.

Graphing Equations

Graphing these equations helps visualize what's happening with their solutions. Graph the equation \( x - y = 3 \). It's a straight line with a slope of 1, meaning it goes up by 1 along the x-axis for every 1 unit it goes up on the y-axis.

Next, graph \( x - y = n/4 \) for the values of \( n \) that we've considered, such as 12 and 10. When \( n = 12 \), the equation simplifies to \( x - y = 3 \), which is the same as the first equation, showing one line on top of another—this is where you see the infinitely many solutions.

For \( n = 10 \), the equation becomes \( x - y = 2.5 \). Though similar in their slopes, they never touch—showing the concept of parallel lines distinctly. Use graphing to understand these relationships better, as visualizing can be the key to understanding complicated concepts in linear equations and their solutions.