Solving Systems Of Equations: Consistent Or Inconsistent?

Hey guys! Let's dive into the fascinating world of systems of equations. Specifically, we're going to figure out how to classify a system and what those classifications actually mean. So, grab your pencils, and let's get started!

Understanding Systems of Equations

Systems of equations are sets of two or more equations that share the same variables. The goal is usually to find values for those variables that satisfy all equations simultaneously. There are several methods to solve these systems, including substitution, elimination, and graphing. But before we even start solving, it’s super helpful to understand the nature of the system itself.

Consistent and Independent Systems

Let's kick things off with consistent and independent systems. A system is considered consistent if it has at least one solution. This means there's at least one set of values for the variables that make all the equations true. Now, what about independent? An independent system has a unique solution. Think of it this way: if you were to graph the equations, they would intersect at exactly one point. This point represents the single, unique solution to the system.

To identify a consistent and independent system, you can try solving it using methods like substitution or elimination. If you arrive at a unique solution for each variable, then you've got yourself a consistent and independent system. Graphically, this means the lines representing the equations intersect at only one point, confirming the single, unique solution. Consider this a classic and straightforward scenario in the world of linear equations.

Consistent and Dependent Systems

Next up, we have consistent and dependent systems. The "consistent" part is the same as before: the system has at least one solution. However, the "dependent" part means that the equations are essentially multiples of each other. In other words, they represent the same line. If you were to graph them, you'd see just one line because the two equations overlap perfectly. Because they're the same line, there are infinitely many solutions. Any point on the line satisfies both equations.

Spotting a consistent and dependent system often involves manipulating the equations. If you can multiply one equation by a constant and get the other equation, you’re dealing with a dependent system. For instance, if you have x + y = 2 and 2x + 2y = 4, you can see that the second equation is just the first equation multiplied by 2. Therefore, they represent the same line and have infinitely many solutions. This type of system might seem tricky, but recognizing the proportional relationship between the equations simplifies the process.

Inconsistent Systems

Last but not least, we have inconsistent systems. An inconsistent system has no solution. This means there's no set of values for the variables that can satisfy all equations simultaneously. Graphically, this means the lines are parallel and never intersect. They have the same slope but different y-intercepts, ensuring they never meet.

Identifying an inconsistent system typically involves trying to solve it. If, during the solving process (using substitution or elimination), you arrive at a contradiction (e.g., 0 = 1), then the system is inconsistent. Another way to identify it is by converting the equations to slope-intercept form (y = mx + b) and noticing that the slopes are the same but the y-intercepts are different. This directly indicates parallel lines and no solution. Inconsistent systems highlight scenarios where the equations contradict each other, making it impossible to find a common solution.

Analyzing the Given System

Now, let's apply this knowledge to the system of equations you provided:

$$\begin{array}{l}7 x-8 y=5 \\ -3 x+16 y=3\end{array}$$

To determine the nature of this system, we can use either substitution or elimination. Let's use elimination.

Elimination Method

Our goal is to eliminate one of the variables by making the coefficients of either x or y opposites. Let's eliminate x. To do this, we can multiply the first equation by 3 and the second equation by 7:

First equation multiplied by 3:

$$3 * (7x - 8y) = 3 * 5 \\ 21x - 24y = 15$$

Second equation multiplied by 7:

$$7 * (-3x + 16y) = 7 * 3 \\ -21x + 112y = 21$$

Now, add the two equations:

$$(21x - 24y) + (-21x + 112y) = 15 + 21 \\ 88y = 36$$

Solve for y:

$$y = \frac{36}{88} = \frac{9}{22}$$

Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation:

$$7x - 8y = 5 \\ 7x - 8(\frac{9}{22}) = 5 \\ 7x - \frac{72}{22} = 5 \\ 7x = 5 + \frac{36}{11} \\ 7x = \frac{55}{11} + \frac{36}{11} \\ 7x = \frac{91}{11} \\ x = \frac{91}{77} = \frac{13}{11}$$

So, we have a unique solution: x = 13/11 and y = 9/22. Since we found a unique solution, the system is consistent and independent.

Conclusion

Therefore, the system of equations is consistent and independent. This means the two lines intersect at a single point, giving us one unique solution. Understanding these classifications helps you anticipate the nature of solutions even before you fully solve the system. Keep practicing, and you'll become a system-solving pro in no time!