Solving Systems Of Equations No Solution Example

Hey guys! Let's dive into solving this system of equations. It looks a bit intimidating at first, but we'll break it down step by step. We've got three equations here, each with three variables: x, y, and z. Our goal is to find the values of these variables that satisfy all three equations simultaneously.

4x - 4y + 5z = -6
12x - 12y + 15z = 9
2x - 4y - 4z = 4

Understanding Systems of Equations

Before we jump into the solution, let's quickly recap what a system of equations is. Essentially, it's a set of two or more equations that share the same variables. The solution to the system is the set of values for the variables that make all the equations true. Think of it like finding the intersection point of multiple lines or planes – the point where they all meet. When you are trying to solve the system of equations, you are looking for this common solution.

There are several methods to tackle systems of equations, including substitution, elimination, and matrix methods. We'll primarily use the elimination method here, as it's quite efficient for this particular system. This method involves manipulating the equations to eliminate one variable at a time, making it easier to solve for the others. The key is to add or subtract multiples of equations to cancel out variables. This is a fundamental technique in linear algebra, and mastering it will greatly help in solving more complex problems.

Step-by-Step Solution

1. Simplify the Equations

First things first, let's see if we can simplify any of the equations. Notice that the second equation (12x - 12y + 15z = 9) is a multiple of the first equation (4x - 4y + 5z = -6). If we multiply the first equation by 3, we get 12x - 12y + 15z = -18. This is interesting because it contradicts the second equation, which states that 12x - 12y + 15z = 9. This contradiction is a big clue!

So, let's multiply the first equation by 3:

3 * (4x - 4y + 5z) = 3 * (-6)
12x - 12y + 15z = -18

Now we have:

12x - 12y + 15z = -18
12x - 12y + 15z = 9

2. Identify the Contradiction

We can clearly see that the left-hand sides of these two equations are identical (12x - 12y + 15z), but the right-hand sides are different (-18 and 9). This means there's no possible combination of x, y, and z that can satisfy both equations simultaneously. This situation indicates that the system of equations is inconsistent, and there is no solution.

Think of it like this: imagine two parallel lines. They never intersect, so there's no point that lies on both lines. Similarly, in this system, the equations represent planes in 3D space that do not intersect at a common point. Therefore, there is no solution.

3. Confirm with Another Pair of Equations (Optional but Recommended)

To be absolutely sure, let's consider another pair of equations. We'll stick with the first equation (4x - 4y + 5z = -6) and the third equation (2x - 4y - 4z = 4). To eliminate x, we can multiply the third equation by -2:

-2 * (2x - 4y - 4z) = -2 * (4)
-4x + 8y + 8z = -8

Now we add this modified equation to the first equation:

(4x - 4y + 5z) + (-4x + 8y + 8z) = -6 + (-8)
4y + 13z = -14

This gives us a new equation: 4y + 13z = -14. While this equation by itself doesn't tell us there's no solution, the earlier contradiction is sufficient evidence. However, going through this exercise reinforces our understanding of the system and the relationships between the equations.

4. State the Conclusion

Since we've identified a contradiction between the first two equations, we can confidently conclude that the system of equations has no solution. There are no values for x, y, and z that will satisfy all three equations simultaneously. This is a crucial understanding because it saves us from wasting time trying to find a solution that doesn't exist.

Why No Solution?

The reason this system has no solution boils down to the geometric interpretation of these equations. Each linear equation in three variables represents a plane in 3D space. When we solve a system of three equations, we're essentially looking for the point (or points) where these three planes intersect. There are a few possibilities:

1. One Unique Solution: The planes intersect at a single point.
2. Infinitely Many Solutions: The planes intersect along a line, or they are the same plane.
3. No Solution: The planes do not have a common intersection point. This can happen if the planes are parallel, or if they intersect in pairs but not all three at once.

In our case, the first two equations represent parallel planes (or planes that would be parallel if they weren't inconsistent), so they never intersect. This geometric perspective helps to visualize why some systems have solutions and others don't.

Common Mistakes to Avoid

When solving systems of equations, there are a few common pitfalls to watch out for:

• Arithmetic Errors: A simple mistake in addition, subtraction, multiplication, or division can throw off the entire solution. Always double-check your calculations!
• Incorrectly Applying Operations: Make sure you're applying operations (like multiplying an equation by a constant) to both sides of the equation.
• Misinterpreting Contradictions: As we saw in this problem, a contradiction indicates no solution. Don't keep trying to solve if you find a contradiction!
• Forgetting to Check Your Solution: If you do find a solution, plug it back into the original equations to make sure it works. This is a great way to catch errors.

Alternative Methods (Briefly Mentioned)

While we primarily used elimination here, let's briefly touch on other methods you could use:

• Substitution: Solve one equation for one variable, then substitute that expression into the other equations. This can be useful when one equation is easily solved for a variable.
• Matrix Methods: Represent the system as a matrix equation and use techniques like Gaussian elimination or finding the inverse of a matrix to solve for the variables. This is particularly useful for larger systems of equations.
• Graphical Methods: For systems of two variables, you can graph the equations and find the intersection point(s). This method provides a visual representation of the solution.

Conclusion: No Solution Found

In summary, after carefully analyzing the given system of equations, we've discovered a contradiction between the first two equations. This contradiction definitively proves that there is no solution to the system. The planes represented by these equations do not intersect at a common point.

So, the correct choice is: There is no solution.

Keep practicing with different systems of equations, guys, and you'll become a pro in no time! Remember to always look for simplifications and contradictions, and don't be afraid to use different methods to find the solution (or determine that one doesn't exist!).