Solving For Y Value In A System Of Linear Equations

When dealing with systems of linear equations, a common task is to find the solution, which represents the point where the lines intersect on a graph. This point is defined by both an $x$-value and a $y$-value that satisfy both equations simultaneously. In this article, we'll explore how to solve a system of linear equations and specifically focus on determining the $y$-value of the solution. We will break down the steps with a concrete example.

Let's consider the following system of linear equations:

$egin{aligned} 4x + 5y &= -12 \ -2x + 3y &= -16

\end{aligned}$

Our goal is to find the $y$-value that satisfies both of these equations. There are several methods to solve such systems, including substitution, elimination, and graphical methods. Here, we will focus on the elimination method, which is particularly effective for this type of problem. The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated, leaving us with a single equation in one variable.

Step 1: Prepare the Equations for Elimination

The elimination method requires us to manipulate the equations so that the coefficients of either $x$ or $y$ are additive inverses (i.e., they add up to zero). Looking at our system,

$egin{aligned} 4x + 5y &= -12 \ -2x + 3y &= -16

\end{aligned}$

we can see that the coefficients of $x$ are $4$ and $-2$. To eliminate $x$, we can multiply the second equation by $2$ so that the coefficients of $x$ become $4$ and $-4$. This will allow us to eliminate $x$ when we add the equations together.

Multiply the second equation by $2$:

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

This simplifies to:

$-4x + 6y = -32$

Now our system of equations looks like this:

$egin{aligned} 4x + 5y &= -12 \ -4x + 6y &= -32

\end{aligned}$

Step 2: Eliminate $x$ by Adding the Equations

Now that the coefficients of $x$ are additive inverses ($4$ and $-4$), we can add the two equations together. This will eliminate $x$ and leave us with an equation in terms of $y$ only.

Adding the equations:

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

This simplifies to:

$11y = -44$

Step 3: Solve for $y$

Now we have a simple equation with just one variable, $y$. To solve for $y$, we divide both sides of the equation by $11$:

y = rac{-44}{11}

$y = -4$

Thus, the $y$-value of the solution is $-4$.

Step 1: Solve One Equation for $x$

To use the substitution method, we first need to solve one of the equations for one variable in terms of the other. Let's solve the second equation for $x$:

$-2x + 3y = -16$

Add $2x$ to both sides:

$3y = 2x - 16$

Add 16 to both sides:

$3y + 16 = 2x$

Divide by 2:

x = rac{3y + 16}{2}

Step 2: Substitute into the Other Equation

Now we substitute this expression for $x$ into the first equation:

$4x + 5y = -12$

Replace $x$ with rac{3y + 16}{2}:

4igg(rac{3y + 16}{2}igg) + 5y = -12

Simplify:

$2(3y + 16) + 5y = -12$

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

$11y + 32 = -12$

Step 3: Solve for $y$

Subtract 32 from both sides:

$11y = -44$

Divide by 11:

$y = -4$

Again, we find that the $y$-value of the solution is $-4$.

From both the elimination and substitution methods, we found that the $y$-value of the solution to the system of equations is $-4$. Looking at the multiple-choice options, we see that:

• A. 5
• B. 2
• C. -2
• D. -4

The correct answer is D. -4.

To ensure our solution is correct, we can substitute the value of $y$ back into the original equations and solve for $x$. Then, we can verify that both $x$ and $y$ values satisfy both equations.

Using the first equation:

$4x + 5y = -12$

Substitute $y = -4$:

$4x + 5(-4) = -12$

$4x - 20 = -12$

Add 20 to both sides:

$4x = 8$

Divide by 4:

$x = 2$

Now, let's check the second equation:

$-2x + 3y = -16$

Substitute $x = 2$ and $y = -4$:

$-2(2) + 3(-4) = -16$

$-4 - 12 = -16$

$-16 = -16$

Since both equations are satisfied, our solution is correct. The solution to the system of equations is $(2, -4)$.

Understanding how to solve systems of linear equations is not just an academic exercise; it has numerous practical applications in various fields. Here are some examples:

1. Economics: In economics, systems of equations are used to model supply and demand curves. The point of intersection of these curves gives the equilibrium price and quantity in a market. For instance, if we have two equations representing the supply and demand for a product, solving the system will tell us the price at which the quantity supplied equals the quantity demanded.

2. Engineering: Engineers use systems of equations to analyze circuits, structural designs, and control systems. For example, in electrical engineering, Kirchhoff's laws can be formulated as a system of linear equations to determine the currents in different branches of a circuit.

3. Computer Graphics: In computer graphics, linear systems are used for transformations, such as scaling, rotation, and translation of objects in 2D and 3D space. These transformations are often represented as matrices, and solving systems of equations involving these matrices is crucial for rendering images.

4. Data Analysis and Statistics: Linear regression, a fundamental tool in statistics, involves solving systems of linear equations to find the best-fit line or curve through a set of data points. This is used in various applications, such as predicting trends, analyzing relationships between variables, and making forecasts.

5. Navigation: GPS systems rely on solving systems of equations to determine the location of a receiver based on signals from multiple satellites. Each satellite provides an equation representing the distance between the receiver and the satellite, and solving this system gives the receiver's coordinates.

6. Resource Allocation: Businesses and governments use linear programming, which involves solving systems of linear inequalities and equations, to optimize resource allocation. This can include determining the most efficient way to distribute goods, schedule employees, or allocate budgets.

7. Chemistry: In chemistry, systems of equations are used in stoichiometry to balance chemical reactions and calculate the amounts of reactants and products involved. For example, balancing a complex chemical equation often requires solving a system of linear equations.

8. Environmental Science: Systems of equations are used to model environmental processes, such as the flow of pollutants in a river or the spread of diseases in a population. These models help in understanding and managing environmental issues.

9. Game Development: In game development, linear algebra and systems of equations are essential for handling game physics, artificial intelligence, and rendering. For instance, systems of equations can be used to calculate trajectories of projectiles, simulate collisions, and control character movements.

10. Cryptography: Cryptographic algorithms often involve solving systems of equations in finite fields. These systems are used for encoding and decoding messages, ensuring secure communication.

These examples illustrate the broad applicability of solving systems of linear equations. The ability to solve these systems is a valuable skill in many fields, making it an essential topic in mathematics education.

In this article, we walked through the process of finding the $y$-value in the solution to a system of linear equations. We demonstrated the elimination and substitution methods, both of which led us to the correct answer of $y = -4$. Additionally, we verified our solution by substituting the $y$-value back into the original equations and solving for $x$. Understanding these methods is crucial for solving various mathematical problems and has wide-ranging applications in real-world scenarios. Mastering these techniques will provide a solid foundation for more advanced mathematical concepts and practical problem-solving situations.