Solving Systems Of Equations: Is (-9,-6) A Solution?

Dec 2, 2025 by Alex Johnson 53 views

In mathematics, especially in algebra, we often encounter systems of equations. These systems consist of two or more equations with the same variables, and our goal is to find the values of these variables that satisfy all equations simultaneously. One common task is to determine whether a given point is a solution to a system of equations. In this article, we'll walk through the process of verifying if the point (-9, -6) is a solution to the following system of equations:

y = (1/3)x - 3
y = -(1/3)x - 5

We will explore the step-by-step method to substitute the given point into the equations and check if both equations hold true. This approach is fundamental in solving and understanding systems of equations, and mastering it will greatly enhance your algebraic skills. Let’s dive in and see how it’s done!

Understanding Systems of Equations

To effectively determine whether a point is a solution to a system of equations, it’s crucial to first grasp what a system of equations truly represents. At its core, a system of equations is a set of two or more equations that share the same variables. The solution to such a system is a set of values for the variables that satisfy all equations simultaneously. Graphically, this solution represents the point(s) where the lines or curves described by the equations intersect. Think of it as finding the specific location where multiple paths cross each other on a map. In our case, we have a system of two linear equations:

y = (1/3)x - 3
y = -(1/3)x - 5

Each of these equations represents a straight line on a coordinate plane. The solution to this system will be the point (x, y) where these two lines intersect. To check if a given point, like (-9, -6), is a solution, we need to verify if substituting x = -9 and y = -6 into both equations makes them true. This process involves basic algebraic substitution and simplification. It's like checking if the coordinates of a specific address match the requirements of multiple routes simultaneously. If the coordinates satisfy all routes, then the address is a valid destination. Understanding this fundamental concept is key to solving more complex systems of equations and related problems in algebra.

Step-by-Step Verification Process

To verify if the point (-9, -6) is a solution to our system of equations, we need to follow a step-by-step process. This involves substituting the x and y values of the point into each equation and checking if the equations hold true. Here’s how we do it:

Identify the Equations:

First, let’s restate the system of equations we are working with:
```
y = (1/3)x - 3
y = -(1/3)x - 5
```
These are the two equations we need to verify.
Substitute the Point (-9, -6) into the First Equation:

We substitute x = -9 and y = -6 into the first equation:
```
-6 = (1/3)(-9) - 3
```
Simplify the First Equation:

Now, let’s simplify the equation:
```
-6 = -3 - 3
-6 = -6
```
Since the equation holds true, the point (-9, -6) satisfies the first equation.
Substitute the Point (-9, -6) into the Second Equation:

Next, we substitute x = -9 and y = -6 into the second equation:
```
-6 = -(1/3)(-9) - 5
```
Simplify the Second Equation:

Let’s simplify this equation as well:
```
-6 = 3 - 5
-6 = -2
```
In this case, the equation does not hold true.
Conclusion:

Since the point (-9, -6) satisfies the first equation but not the second, it is not a solution to the system of equations. For a point to be a solution, it must satisfy all equations in the system. This methodical approach ensures accuracy and clarity in determining the solutions to systems of equations. It’s like checking if a key fits all the locks in a set; if it only fits some, it’s not the right key for the entire set.

Detailed Analysis of the First Equation

Let's delve deeper into the analysis of the first equation to understand how we confirmed whether the point (-9, -6) satisfies it. The first equation in our system is:

y = (1/3)x - 3

To check if (-9, -6) is a solution, we substitute x = -9 and y = -6 into this equation. This gives us:

-6 = (1/3)(-9) - 3

The next step is to simplify the right-hand side of the equation. We start by multiplying (1/3) by -9:

(1/3) * -9 = -3

Now, we substitute this result back into the equation:

-6 = -3 - 3

Next, we perform the subtraction:

-3 - 3 = -6

So, our equation becomes:

-6 = -6

This statement is true, which means that the point (-9, -6) does indeed satisfy the first equation. Graphically, this implies that the point (-9, -6) lies on the line represented by the equation y = (1/3)x - 3. It’s essential to go through each step meticulously to ensure accuracy, as even a small error in calculation can lead to a wrong conclusion. This process of substituting and simplifying is a fundamental technique in algebra, especially when dealing with linear equations and systems of equations. It’s like verifying if a set of coordinates aligns perfectly with the route specified by an equation on a graph.

Detailed Analysis of the Second Equation

Now, let’s conduct a detailed analysis of the second equation to see why the point (-9, -6) does not satisfy it. The second equation in our system is:

y = -(1/3)x - 5

Similar to the first equation, we substitute x = -9 and y = -6 into this equation:

-6 = -(1/3)(-9) - 5

The first step in simplifying the right-hand side is to multiply -(1/3) by -9. Remember that a negative times a negative results in a positive:

-(1/3) * -9 = 3

Now, we substitute this result back into the equation:

-6 = 3 - 5

Next, we perform the subtraction:

3 - 5 = -2

So, our equation simplifies to:

-6 = -2

This statement is false. The left side, -6, is not equal to the right side, -2. This indicates that the point (-9, -6) does not satisfy the second equation. Graphically, this means that the point (-9, -6) does not lie on the line represented by the equation y = -(1/3)x - 5. This detailed breakdown highlights the importance of careful calculation and accurate arithmetic when verifying solutions to equations. The point must satisfy every equation in the system to be considered a valid solution for the entire system. In this case, the failure of the point to satisfy the second equation is enough to disqualify it as a solution to the system.

Conclusion: Why (-9, -6) Is Not a Solution

In conclusion, we’ve demonstrated that the point (-9, -6) is not a solution to the given system of equations:

y = (1/3)x - 3
y = -(1/3)x - 5

We arrived at this conclusion by systematically substituting the x and y values of the point into each equation and verifying whether the equations hold true. The point (-9, -6) satisfies the first equation, y = (1/3)x - 3, as we showed that -6 = (1/3)(-9) - 3 simplifies to -6 = -6, which is a true statement. However, when we substituted (-9, -6) into the second equation, y = -(1/3)x - 5, we found that -6 = -(1/3)(-9) - 5 simplifies to -6 = -2, which is a false statement. For a point to be a solution to a system of equations, it must satisfy all equations in the system simultaneously. Since (-9, -6) only satisfies the first equation but not the second, it is not a solution to the system. This methodical approach of substitution and simplification is a fundamental technique in algebra. It’s crucial for accurately determining the solutions to systems of equations and for understanding the graphical representations of these solutions. Each point must lie on all lines (or curves) defined by the equations in the system to be considered a valid solution.

For further exploration of systems of equations, you might find valuable resources and explanations on websites like Khan Academy's Systems of Equations Section, which offers comprehensive lessons and practice exercises.