Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether each system of linear equations has (a) one and only one solution, (b) infinitely many solutions, or (c) no solution. Find all solutions whenever they exist. \(\begin{aligned} 3 x-4 y &=7 \\ 9 x-12 y &=14 \end{aligned}\)

Step-by-Step Solution

Answer

The given system of linear equations has \(\textbf{(c) no solution}\), as both equations are parallel and will never intersect.

Step 1: Write the given system of linear equations

We are given the following system: \(\begin{cases} 3x-4y = 7 \\ 9x-12y = 14 \end{cases}\)

Step 2: Perform elimination method

To eliminate one of the variables, we need to make the coefficients of either \(x\) or \(y\) the same in both equations. In this case, the coefficients of \(x\) can be made the same by multiplying the first equation by 3. This would give us: \[\begin{cases} 3(3x-4y) = 3(7) \\ 9x-12y = 14 \end{cases}\] Simplifying, we get: \(\begin{cases} 9x-12y = 21 \\ 9x-12y = 14 \end{cases}\)

Step 3: Observe the coefficients and constants

We can see that both equations have the same coefficients for \(x\) and \(y\), but different constants on the right-hand side. Therefore, these equations are parallel and will never intersect. Since the equations never intersect, there is no point where both equations are satisfied simultaneously, which means there is #c#no solution# for this system of linear equations.

Key Concepts

Elimination Method

The elimination method is a staple in solving systems of linear equations. It involves manipulating the equations so you can eliminate one variable and solve for the other. This is particularly useful for systems with two equations and two unknowns, but can be scaled to larger systems as well. Here’s how it generally works:

• Begin by aligning your equations and looking for a variable that can be eliminated by addition or subtraction.
• Manipulate either one or both of the equations by multiplication or division so that the coefficients of the chosen variable are opposites or the same.
• Add or subtract the equations from each other to eliminate the chosen variable.
• Once a variable is eliminated, you are left with an equation in one variable, which you can solve.
• Substitute the solved value back into one of the original equations to find the value of the other variable.

Remember, the goal is to reduce a system of multiple equations down to one that can be easily solved. This is precisely what was attempted in the exercise, targeting the variable x for elimination by trying to match its coefficients.

Parallel Lines

In the context of linear equations, parallel lines represent a pair of lines that never intersect, no matter how far they are extended. This concept translates into mathematics as two linear equations with identical slopes but different y-intercepts.

Why does this matter? When we're solving systems of linear equations, we're looking for a point or points where the two lines, or more, intersect, which represents the solution to the system. If the lines are parallel, this point of intersection does not exist. When lines are graphed on a coordinate plane, the slope of the line is determined by the coefficient of x when the equation is in the slope-intercept form, y = mx + b. If two lines have the same m value, but different b values, they will never meet, which is why in the given exercise, we can quickly determine there is an issue just by examining the equations and noticing they have the same coefficients for x and y.

No Solution

When we encounter the term 'no solution' in the world of systems of linear equations, it means that there’s no set of values for our variables that makes all the equations true simultaneously. A system with no solution is also called an inconsistent system. It can occur in a couple of different scenarios:

• The most common is when we have parallel lines, as we saw in the exercise. The lines never intersect, implying that there's no point that exists on both lines.
• Another scenario could be when the same coefficient is multiplied by a constant, but the results do not align, suggesting no possible equality can be obtained.

In our exercise, after applying the elimination method, we ended up with two equations that looked identical in terms of variables but had different constants, conclusively introducing us to a situation with no solution. Recognizing such cases saves time and effort, redirecting the focus on understanding the implications rather than fruitlessly trying to solve an unsolvable system.