Question

Perform parts (a)-(c). 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. $$ \begin{aligned}&9 x+6 y=n\\\&1.8 x+1.2 y=3\end{aligned} $$

Short answer

For the system to have infinitely many solutions, \(n = 15\). For the system to have no solution, \(n\) must be any value other than 15.

Step-by-step solution

Finding a Value of \(n\) for Infinitely Many Solutions

To obtain infinitely many solutions, the two equations must correspond to the same line. This means that they should be proportional to each other. Therefore, the ratios of the coefficients of \(x\), \(y\), and the constants should all be equal. Hence, \(9/1.8 = 6/1.2 = n/3\). Solving this gives \(n = 15\).

Finding a value of \(n\) for No Solution

No solution occurs if the equations refer to parallel lines, which means they have the same coefficients of \(x\) and \(y\) but different constants. Thus, \(9/1.8 = 6/1.2 \neq n/3\). You solve this inequality for \(n\). For each given value of \(n\neq15\), there will be no solution for the system of linear equations.

Graphing the Results

For the graph, first, the two given lines need to be plotted. For \(9x + 6y = n\), when \(n=15\), that's the same line as \(1.8x + 1.2y = 3\). So, this line will be plotted once, showing it has infinitely many solutions. For the same line when \(n\neq15\), plot a line parallel to the previous one, which represents there are no solutions as they never intersect.

Key concepts

Infinitely Many Solutions

When we talk about a system of linear equations, we usually expect to find a unique solution where the two lines intersect at a single point. However, there are cases where the system may have 'infinitely many solutions'. This happens when the two equations represent the exact same line, which means every point on the line is a solution to the system.

For instance, if we have two equations, let's say, \(9x + 6y = n\) and \(1.8x + 1.2y = 3\), to find a value of \(n\) that results in infinitely many solutions, we look for a proportional constant that makes one equation a scalar multiple of the other. In simple terms, we need to make sure that the ratio of the coefficients of \(x\), the coefficients of \(y\), and the constant terms on the other side of the equation are identical. Solving the proportional relationship \(9/1.8 = 6/1.2 = n/3\) gives us the magical value of \(n = 15\), whereupon the two equations describe the same line, and thus any point on that line is a valid solution.

By graphing the equations, we would see just one line instead of two intersecting lines, which visually confirms that there are infinitely many solutions. It's like having a road that goes on forever, with countless spots to stop - except in this case, each 'stop' is a solution!

No Solution Systems

In sharp contrast to systems with infinitely many solutions are those systems that offer 'no solution'. This situation arises when we have two lines that are parallel to each other; since they never meet, there is no point that lies on both lines simultaneously.

In terms of our example with equations \(9x + 6y = n\) and \(1.8x + 1.2y = 3\), a 'no solution' scenario occurs when we select a value for \(n\) that makes the lines parallel. We already know that if \(n = 15\), the equations describe the same line, but what if \(n\) is anything but 15? That's when we'd get parallel lines. These lines have the same slope (which means the ratio of the coefficients of \(x\) and \(y\) are equal, such as \(9/1.8 = 6/1.2\)) but different y-intercepts (since \(n\) isn't 15, the constant term changes, preventing the equal ratio).

By setting \(n\) to any value other than 15, we ensure that while the lines move up and down along the y-axis, they'll do so without ever crossing paths. It's like two tracks running side by side without meeting - symbolizing a journey of isolation with no shared destinations.

Graphing Linear Equations

To truly understand a system of linear equations, nothing beats the power of visualization, and that's where 'graphing linear equations' comes in. This process involves taking algebraic equations and representing them visually on a coordinate plane, making it easier to identify solutions.

Graphing is like drawing a roadmap for the equations at hand. For our equations \(9x + 6y = n\) and \(1.8x + 1.2y = 3\), graphing involves plotting points that satisfy each equation and then connecting these points to form lines. When \(n=15\), the equation \(9x + 6y = n\) becomes identical to \(1.8x + 1.2y = 3\), and so they are graphed as the same line. Conversely, with a different value for \(n\), the line \(9x + 6y = n\) shifts parallel to the original, without touching it, illustrating a 'no solution' case.

To effectively graph a line, we usually start by plotting the y-intercept (where the line crosses the y-axis) and then use the slope to find another point. Connecting these points with a straight line unveils the path that the equation carves through the coordinate plane, a real-life representation of how math meets the eye!