Solving Systems Of Linear Equations Finding The Ordered Pair Solution

In the realm of mathematics, particularly in algebra, solving systems of linear equations is a fundamental skill. These systems arise in various real-world applications, from economics and engineering to computer science and data analysis. Understanding how to find the solution to a system of equations is crucial for making informed decisions and predictions. In this article, we will delve into the process of solving a specific system of linear equations to find the ordered pair extbf{solution}, providing a step-by-step guide and clear explanations to enhance your understanding.

Understanding Systems of Linear Equations

To effectively tackle the problem at hand, it's essential to first grasp the concept of systems of linear equations. A system of linear equations consists of two or more linear equations involving the same variables. The solution to the system is the set of values for the variables that satisfy all the equations simultaneously. Graphically, the solution represents the point(s) where the lines corresponding to the equations intersect.

There are several methods to solve systems of linear equations, including:

• Substitution: Solving one equation for one variable and substituting that expression into the other equation(s).
• Elimination: Adding or subtracting multiples of the equations to eliminate one variable.
• Graphing: Plotting the equations and finding the point(s) of intersection.
• Matrix Methods: Using matrices and their properties to solve the system, especially useful for larger systems.

For the given problem, we will primarily focus on the elimination method, as it offers a straightforward approach for this particular system.

The Given System of Equations

We are presented with the following system of linear equations:

$$\left\{ \begin{aligned} 3 a+b & =10 \\ -4 a-2 b & =2 \end{aligned} \right.$$

Our goal is to find the ordered pair ($a$, $b$) that satisfies both equations simultaneously. This means we need to find values for $a$ and $b$ that, when substituted into both equations, make the equations true.

Solving the System Using Elimination

The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated. To achieve this, we can multiply one or both equations by constants so that the coefficients of one variable are opposites. In our case, we can multiply the first equation by 2 to make the coefficient of $b$ equal to 2, which is the opposite of the coefficient of $b$ in the second equation.

Step 1: Multiply the First Equation by 2

Multiplying the first equation ($3a + b = 10$) by 2, we get:

$$2(3a + b) = 2(10)$$

$$6a + 2b = 20$$

Now our system of equations looks like this:

$$\left\{ \begin{aligned} 6a + 2b & = 20 \\ -4a - 2b & = 2 \end{aligned} \right.$$

Step 2: Add the Modified Equations

Next, we add the two equations together:

$$(6a + 2b) + (-4a - 2b) = 20 + 2$$

Combining like terms, we have:

$$6a - 4a + 2b - 2b = 22$$

$$2a = 22$$

Notice that the $b$ terms have canceled out, leaving us with a single equation in terms of $a$.

Step 3: Solve for $a$

To find the value of $a$, we divide both sides of the equation by 2:

$$\frac{2a}{2} = \frac{22}{2}$$

$$a = 11$$

So, we have found that $a = 11$.

Step 4: Substitute the Value of $a$ into One of the Original Equations

Now that we have the value of $a$, we can substitute it into either of the original equations to solve for $b$. Let's use the first equation, $3a + b = 10$:

$$3(11) + b = 10$$

$$33 + b = 10$$

Step 5: Solve for $b$

To solve for $b$, we subtract 33 from both sides of the equation:

$$33 + b - 33 = 10 - 33$$

$$b = -23$$

Thus, we have found that $b = -23$.

Step 6: Write the Solution as an Ordered Pair

We have determined that $a = 11$ and $b = -23$. Therefore, the solution to the system of equations is the ordered pair $(11, -23)$.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the values of $a$ and $b$ back into both original equations.

Verification in the First Equation

Substituting $a = 11$ and $b = -23$ into $3a + b = 10$:

$$3(11) + (-23) = 10$$

$$33 - 23 = 10$$

$$10 = 10$$

The first equation holds true.

Verification in the Second Equation

Substituting $a = 11$ and $b = -23$ into $-4a - 2b = 2$:

$$-4(11) - 2(-23) = 2$$

$$-44 + 46 = 2$$

$$2 = 2$$

The second equation also holds true. Since the values of $a$ and $b$ satisfy both equations, our solution is correct.

Analyzing the Answer Choices

Now that we have found the solution to be $(11, -23)$, let's examine the given answer choices:

A. $(1, 7)$ B. $(3, 1)$ C. $(11, -23)$ D. $(23, -11)$

We can clearly see that option C, $(11, -23)$, matches our solution.

Conclusion

In this article, we successfully solved the given system of linear equations using the elimination method. By carefully manipulating the equations, we eliminated one variable and solved for the other. We then substituted the value back into one of the original equations to find the remaining variable. Finally, we verified our solution by plugging the values back into both equations and confirming their validity. The ordered pair solution to the system is $(11, -23)$, which corresponds to option C.

Understanding and applying these methods for solving systems of linear equations is crucial for various mathematical and real-world problems. Whether it's determining the break-even point for a business, calculating the trajectory of a projectile, or analyzing data trends, the ability to solve these systems is an invaluable skill.

By mastering the techniques discussed, you'll be well-equipped to tackle more complex problems and gain a deeper understanding of the mathematical principles that govern our world. Remember, practice is key to proficiency, so continue to work through different examples and challenge yourself to expand your problem-solving abilities. Solving systems of equations is a fundamental concept in mathematics with vast applications in various fields. This article has provided a comprehensive guide on how to solve a specific system and, more broadly, how to approach such problems. Whether you're a student learning the basics or a professional applying these concepts in your work, the skills and knowledge gained here will serve you well.