Solving Systems Of Linear Equations Finding The Ordered Pair Solution

by ADMIN 70 views

In mathematics, particularly in algebra, solving systems of linear equations is a fundamental skill. This article aims to provide a comprehensive guide on how to solve such systems, using the given example as a practical illustration. We will explore various methods to find the ordered pair (a, b) that satisfies both equations simultaneously. Understanding these methods is crucial for students and anyone dealing with mathematical problems in various fields.

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations containing two or more variables. The solution to a system of linear equations is the set of values that, when substituted for the variables, make all the equations true. Graphically, the solution represents the point(s) where the lines corresponding to the equations intersect. If the lines are parallel, there is no solution, and if they are the same line, there are infinitely many solutions. The given system of linear equations is:

3a + b = 10
-4a - 2b = 2

Our goal is to find the values of a and b that satisfy both equations. There are several methods to achieve this, including substitution, elimination, and graphical methods. We will primarily focus on the substitution and elimination methods in this article.

Method 1: The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can be easily solved. Let's apply this method to our system.

Step 1: Solve one equation for one variable

From the first equation, 3a + b = 10, we can easily solve for b:

b = 10 - 3a

Step 2: Substitute the expression into the other equation

Now, substitute this expression for b into the second equation, -4a - 2b = 2:

-4a - 2(10 - 3a) = 2

Step 3: Solve for the remaining variable

Simplify and solve for a:

-4a - 20 + 6a = 2
2a - 20 = 2
2a = 22
a = 11

Step 4: Substitute back to find the other variable

Now that we have a = 11, substitute this value back into the expression for b:

b = 10 - 3(11)
b = 10 - 33
b = -23

Thus, the solution using the substitution method is (a, b) = (11, -23).

Method 2: The Elimination Method

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. When the equations are added, that variable is eliminated, leaving a single equation with one variable. Let's apply the elimination method to our system.

Step 1: Manipulate the equations to have opposite coefficients

Multiply the first equation by 2 to make the coefficient of b equal to 2:

2(3a + b) = 2(10)
6a + 2b = 20

Now we have the system:

6a + 2b = 20
-4a - 2b = 2

Step 2: Add the equations to eliminate a variable

Add the two equations together:

(6a + 2b) + (-4a - 2b) = 20 + 2
2a = 22

Step 3: Solve for the remaining variable

Solve for a:

a = 11

Step 4: Substitute back to find the other variable

Substitute a = 11 into one of the original equations. Let’s use the first equation 3a + b = 10:

3(11) + b = 10
33 + b = 10
b = 10 - 33
b = -23

Thus, the solution using the elimination method is (a, b) = (11, -23).

Verification of the Solution

To ensure our solution is correct, we must verify it by substituting the values of a and b back into both original equations.

Equation 1: 3a + b = 10

Substitute a = 11 and b = -23:

3(11) + (-23) = 10
33 - 23 = 10
10 = 10

The first equation holds true.

Equation 2: -4a - 2b = 2

Substitute a = 11 and b = -23:

-4(11) - 2(-23) = 2
-44 + 46 = 2
2 = 2

The second equation also holds true. Therefore, the ordered pair (11, -23) is indeed the solution to the given system of linear equations.

Analyzing the Options

Now, let's analyze the given options to determine which one matches our solution:

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

By comparing our solution (11, -23) with the options, we can clearly see that option C is the correct answer.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including:

  • Economics: Analyzing supply and demand, determining equilibrium prices, and modeling economic growth.
  • Engineering: Designing structures, analyzing circuits, and modeling systems.
  • Computer Science: Developing algorithms, creating simulations, and optimizing performance.
  • Physics: Describing motion, calculating forces, and modeling physical phenomena.
  • Chemistry: Balancing chemical equations and calculating reaction rates.

For instance, in economics, linear equations can represent the relationship between the price of a product and the quantity demanded. By solving a system of linear equations, economists can determine the market equilibrium, where supply equals demand. In engineering, linear equations are used to analyze the forces acting on a bridge or the flow of electricity in a circuit. The ability to solve systems of linear equations is a valuable skill in many professions.

Common Mistakes and How to Avoid Them

When solving systems of linear equations, several common mistakes can occur. Being aware of these mistakes and knowing how to avoid them can significantly improve your accuracy and efficiency.

Sign Errors

One of the most common mistakes is making sign errors, especially when distributing a negative sign or combining like terms. To avoid this, always double-check your signs and be meticulous in your calculations. Write out each step clearly, and use parentheses to keep track of negative signs.

Arithmetic Errors

Simple arithmetic errors, such as adding or multiplying incorrectly, can also lead to incorrect solutions. Use a calculator to verify your calculations, especially when dealing with larger numbers or complex expressions. Practice mental math to improve your speed and accuracy.

Incorrect Substitution

When using the substitution method, it’s crucial to substitute the expression correctly into the other equation. Ensure that you are substituting for the correct variable and that you are distributing any coefficients properly. Double-check your substitution before proceeding.

Elimination Errors

In the elimination method, a common mistake is failing to multiply all terms in the equation by the same constant. Ensure that you multiply every term, including the constant term, to maintain the equality. Also, be careful when adding or subtracting equations to avoid sign errors.

Not Verifying the Solution

Always verify your solution by substituting the values back into the original equations. This is the best way to catch any mistakes and ensure that your solution is correct. If the solution does not satisfy both equations, you know you have made an error and need to review your work.

Misinterpreting No Solution or Infinite Solutions

Sometimes, a system of linear equations may have no solution (parallel lines) or infinite solutions (the same line). Be able to recognize these cases. If you arrive at a contradiction (e.g., 0 = 1) when solving, the system has no solution. If you arrive at an identity (e.g., 0 = 0), the system has infinite solutions.

By being mindful of these common mistakes and practicing regularly, you can improve your skills in solving systems of linear equations.

Conclusion

In conclusion, solving systems of linear equations is a vital skill in mathematics with broad applications in various fields. We have demonstrated two primary methods – substitution and elimination – to solve the given system:

3a + b = 10
-4a - 2b = 2

Both methods led us to the solution (a, b) = (11, -23), which we verified by substituting the values back into the original equations. Understanding these methods and practicing them will empower you to tackle more complex mathematical problems and real-world scenarios. Remember to avoid common mistakes by carefully checking your work and verifying your solutions. The correct answer to the question, "Which ordered pair (a, b) is the solution to the given system of linear equations?", is C. (11, -23).

By mastering the techniques discussed in this article, you'll be well-equipped to solve any system of linear equations you encounter, making this fundamental algebraic skill a valuable asset in your mathematical toolkit.