Finding The Y-Value Solution For Linear Equations 4x + 5y = -12 And -2x + 3y = -16

Jul 25, 2025 by Admin 83 views

Unveiling the Y-Value Solution in a System of Linear Equations

Navigating the realm of linear equations can sometimes feel like deciphering a secret code, but fear not, math enthusiasts! We're here to break down the process of finding the elusive y-value in a system of linear equations. Specifically, we'll tackle the following system:

4x + 5y = -12
-2x + 3y = -16

Our mission? To determine the y-value that satisfies both equations simultaneously. So, let's put on our detective hats and embark on this mathematical journey together!

Methods to Solve Systems of Linear Equations

Dive into the Substitution Method

Let's start by mastering the substitution method, guys. This technique involves isolating one variable in one equation and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation that we can easily solve. Think of it as a clever way to simplify the problem!

In our case, let's choose the second equation (-2x + 3y = -16) and solve for x. We can do this by adding 2x to both sides and then dividing by 2:

-2x + 3y = -16
3y = 2x - 16
x = (3y + 16) / 2

Now, we have an expression for x in terms of y. The next step is to substitute this expression into the first equation (4x + 5y = -12):

4((3y + 16) / 2) + 5y = -12

Simplify this equation by multiplying out the terms and combining like terms:

6y + 32 + 5y = -12
11y + 32 = -12

Subtract 32 from both sides:

11y = -44

Finally, divide by 11 to find the value of y:

y = -4

And there you have it! We've successfully used the substitution method to find that the y-value is -4. It's like cracking the code, isn't it?

Elimination Method: A Powerful Alternative

Now, let's explore another powerful technique for solving systems of linear equations: the elimination method. This method involves manipulating the equations so that the coefficients of one of the variables are opposites. When we add the equations together, that variable is eliminated, leaving us with a single equation in one variable.

Looking at our system:

4x + 5y = -12
-2x + 3y = -16

We can see that the coefficients of x are 4 and -2. To make them opposites, we can multiply the second equation by 2:

2(-2x + 3y) = 2(-16)
-4x + 6y = -32

Now, we have the following system:

4x + 5y = -12
-4x + 6y = -32

Notice that the coefficients of x are now 4 and -4, which are opposites. We can add the two equations together to eliminate x:

(4x + 5y) + (-4x + 6y) = -12 + (-32)
11y = -44

Divide both sides by 11:

y = -4

Voila! We've arrived at the same answer, y = -4, using the elimination method. It's like having two different keys that unlock the same secret!

Solving the System: A Step-by-Step Walkthrough

To solidify our understanding, let's walk through the solution process step by step, emphasizing the elimination method. Remember, the key is to manipulate the equations to eliminate one variable.

Identify the Target Variable: In our system,
```
4x + 5y = -12
-2x + 3y = -16
```
it seems easier to eliminate x since the coefficients 4 and -2 are readily convertible to opposites. This strategic move simplifies our calculations. By focusing on eliminating x, we streamline the process and reduce the complexity of the system.
Multiply to Create Opposites: To make the x coefficients opposites, multiply the second equation by 2. This gives us:
```
-4x + 6y = -32
```
Now we align this modified equation with the original first equation, setting the stage for the elimination of x. This step is crucial as it prepares the system for the subsequent addition of equations, which will lead us closer to the solution.
Add the Equations: Add the modified second equation to the first equation:
```
(4x + 5y) + (-4x + 6y) = -12 + (-32)
```
This results in:
```
11y = -44
```
The addition process effectively cancels out the x terms, leaving us with a simpler equation in terms of y. This simplification is a significant step forward, bringing us closer to isolating y and determining its value.
Solve for y: Divide both sides by 11:
```
y = -4
```
This step definitively reveals the y-value of the solution. By isolating y, we've achieved a key milestone in solving the system of equations. This value is a critical component of the solution, providing one of the coordinates of the point where the two lines intersect.
Optional: Solve for x: If needed, substitute y = -4 back into either original equation to solve for x. Let’s use the first equation:
```
4x + 5(-4) = -12
4x - 20 = -12
4x = 8
x = 2
```
This optional step completes the solution by determining the corresponding x-value. Substituting the found y-value back into one of the original equations allows us to solve for x, providing a comprehensive solution to the system. The x-value, along with the previously found y-value, gives us the coordinates of the intersection point of the two lines represented by the equations.

The Y-Value Unveiled: -4

Through both the substitution and elimination methods, we've confidently determined that the y-value in the solution to this system of linear equations is -4. This value represents the y-coordinate of the point where the two lines intersect on a graph. It's a crucial piece of information that helps us understand the relationship between these two equations.

Real-World Applications: Why This Matters

Now, you might be wondering,