Solving Systems Of Equations Ordered Pair Solutions (a, B)

Jul 12, 2025 by ADMIN 59 views

Solving Systems of Equations A Step-by-Step Guide to Finding Ordered Pair Solutions

In the realm of mathematics, solving systems of equations is a fundamental skill with wide-ranging applications. From determining the intersection points of lines to modeling real-world scenarios, the ability to find solutions that satisfy multiple equations simultaneously is invaluable. This article will delve into the process of solving systems of equations, specifically focusing on a system of two linear equations with two variables. We will guide you through the steps to arrive at the solution, expressed as an ordered pair in the format (a, b).

Understanding Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. The solution to a system of equations is the set of values for the variables that make all the equations true simultaneously. Graphically, the solution represents the point(s) where the lines or curves represented by the equations intersect. In this article, we will focus on systems of two linear equations with two variables, which represent two straight lines in a coordinate plane.

Linear equations are equations where the variables are raised to the power of 1, and the graph of the equation is a straight line. A system of two linear equations with two variables can have one solution, no solution, or infinitely many solutions. One solution indicates that the lines intersect at one point, no solution indicates that the lines are parallel and do not intersect, and infinitely many solutions indicate that the lines are coincident (the same line).

Methods for Solving Systems of Equations

There are several methods for solving systems of equations, including:

Substitution Method: In this method, we solve one equation for one variable and substitute that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved. The value of this variable is then substituted back into one of the original equations to find the value of the other variable.
Elimination Method: Also known as the addition or subtraction method, this method involves manipulating the equations so that the coefficients of one variable are opposites. By adding the equations, one variable is eliminated, and we are left with a single equation with one variable. Solving this equation and substituting back into one of the original equations yields the solution.
Graphical Method: This method involves graphing the equations and finding the point(s) of intersection. The coordinates of the intersection point(s) represent the solution(s) to the system.
Matrix Method: This method uses matrix operations to solve the system of equations. It is particularly useful for systems with more than two equations and variables.

In this article, we will primarily use the elimination method to solve the given system of equations.

Solving the System of Equations

Let's consider the following system of equations:

3x + 4y = 16
-4x - 3y = -19

Our goal is to find the values of x and y that satisfy both equations. We will use the elimination method to achieve this.

Step 1: Multiplying the Equations

To eliminate one of the variables, we need to make the coefficients of either x or y opposites. Let's eliminate x. To do this, we will multiply the first equation by 4 and the second equation by 3. This will make the coefficients of x in the two equations 12 and -12, respectively.

Multiplying the first equation by 4, we get:

4(3x + 4y) = 4(16)
12x + 16y = 64

Multiplying the second equation by 3, we get:

3(-4x - 3y) = 3(-19)
-12x - 9y = -57

Now our system of equations looks like this:

12x + 16y = 64
-12x - 9y = -57

Step 2: Adding the Equations

Now that the coefficients of x are opposites, we can add the two equations to eliminate x:

(12x + 16y) + (-12x - 9y) = 64 + (-57)
12x + 16y - 12x - 9y = 7
7y = 7

Step 3: Solving for y

We now have a single equation with one variable, y. To solve for y, we divide both sides of the equation by 7:

7y / 7 = 7 / 7
y = 1

So, the value of y is 1.

Step 4: Substituting to Find x

Now that we have the value of y, we can substitute it into either of the original equations to find the value of x. Let's substitute y = 1 into the first equation:

3x + 4y = 16
3x + 4(1) = 16
3x + 4 = 16

Step 5: Solving for x

To solve for x, we first subtract 4 from both sides of the equation:

3x + 4 - 4 = 16 - 4
3x = 12

Then, we divide both sides by 3:

3x / 3 = 12 / 3
x = 4

So, the value of x is 4.

Step 6: Expressing the Solution as an Ordered Pair

The solution to the system of equations is the ordered pair (x, y), which in this case is (4, 1). This means that the values x = 4 and y = 1 satisfy both equations in the system.

Verifying the Solution

To ensure that our solution is correct, we can substitute the values of x and y back into the original equations and check if they hold true.

Substituting x = 4 and y = 1 into the first equation:

3x + 4y = 16
3(4) + 4(1) = 16
12 + 4 = 16
16 = 16

The first equation is satisfied.

Substituting x = 4 and y = 1 into the second equation:

-4x - 3y = -19
-4(4) - 3(1) = -19
-16 - 3 = -19
-19 = -19

The second equation is also satisfied. Therefore, our solution (4, 1) is correct.

Conclusion

In this article, we have demonstrated how to solve a system of two linear equations with two variables using the elimination method. We walked through the steps of multiplying the equations to make the coefficients of one variable opposites, adding the equations to eliminate that variable, solving for the remaining variable, and substituting back to find the value of the eliminated variable. Finally, we expressed the solution as an ordered pair (a, b) and verified our answer by substituting the values back into the original equations.

Solving systems of equations is a crucial skill in mathematics and has numerous applications in various fields. By mastering these techniques, you can confidently tackle a wide range of problems involving multiple equations and variables. Remember to practice regularly and explore different methods to find the approach that works best for you. With dedication and perseverance, you can excel in solving systems of equations and unlock the power of mathematical problem-solving.

This article provides a comprehensive guide to solving systems of equations. By following the steps outlined and practicing consistently, you can develop a strong understanding of this fundamental mathematical concept and apply it to solve real-world problems.

Practice Problems

To further solidify your understanding of solving systems of equations, try solving the following practice problems:

Solve the following system of equations:
```
2x - y = 5
x + 2y = 10
```
Solve the following system of equations:
```
5x + 3y = 15
-2x + 4y = 8
```
Solve the following system of equations:
```
4x - 2y = 6
6x + y = 15
```

Check your answers by substituting the solutions back into the original equations. With practice, you'll become proficient in solving systems of equations.

By understanding the underlying concepts and practicing regularly, you can master the art of solving systems of equations and confidently tackle a wide range of mathematical problems.

This section focuses on how to solve a specific system of equations and express the answer as an ordered pair in the format (a, b). We will revisit the same system of equations used earlier and reinforce the steps involved in finding the solution. Understanding how to express the solution correctly is crucial for clear communication and accurate representation of the answer.

Review of the System of Equations

Let's consider the following system of equations again:

3x + 4y = 16
-4x - 3y = -19

As we previously discussed, our objective is to determine the values of x and y that simultaneously satisfy both equations. We will employ the elimination method, a powerful technique for solving such systems.

Step-by-Step Solution Using Elimination Method

To recap, the elimination method involves manipulating the equations to make the coefficients of one variable opposites. This allows us to eliminate that variable by adding the equations together, resulting in a single equation with one variable. Let's walk through the steps again for clarity.

Step 1: Multiplying the Equations (Revisited)

To eliminate x, we need to make its coefficients opposites. We multiply the first equation by 4 and the second equation by 3:

4(3x + 4y) = 4(16)  =>  12x + 16y = 64
3(-4x - 3y) = 3(-19) => -12x - 9y = -57

Now our system looks like this:

12x + 16y = 64
-12x - 9y = -57

Step 2: Adding the Equations (Revisited)

Adding the equations eliminates x:

(12x + 16y) + (-12x - 9y) = 64 + (-57)
12x + 16y - 12x - 9y = 7
7y = 7

Step 3: Solving for y (Revisited)

Dividing both sides by 7 gives us the value of y:

7y / 7 = 7 / 7
y = 1

Step 4: Substituting to Find x (Revisited)

Substituting y = 1 into the first original equation allows us to solve for x:

3x + 4y = 16
3x + 4(1) = 16
3x + 4 = 16

Step 5: Solving for x (Revisited)

Subtracting 4 from both sides and then dividing by 3 gives us the value of x:

3x = 12
3x / 3 = 12 / 3
x = 4

Step 6: Expressing the Solution as an Ordered Pair (a, b) - Key Focus

This is a crucial step. The solution to a system of two equations with two variables is represented as an ordered pair (a, b), where 'a' is the value of x and 'b' is the value of y. In our case, x = 4 and y = 1.

Therefore, the solution to the system of equations is (4, 1). This ordered pair represents the point of intersection of the two lines represented by the equations in a coordinate plane.

Importance of the Ordered Pair Format

The ordered pair format is essential for several reasons:

Clarity: It clearly indicates the values of x and y in a specific order, ensuring that there is no ambiguity in the solution.
Graphical Representation: It directly corresponds to a point in the coordinate plane, making it easy to visualize the solution.
Consistency: It provides a standard format for expressing solutions to systems of equations, facilitating communication and understanding.

Emphasizing No Spaces in the Ordered Pair

It is crucial to write the ordered pair with no spaces between the numbers or symbols. The correct format is (4, 1), not (4, 1) or ( 4, 1). The absence of spaces ensures that the ordered pair is recognized as a single entity representing a point in the coordinate plane.

Common Mistakes to Avoid

Reversing the Order: Ensure that the x-value comes first and the y-value comes second in the ordered pair. (1, 4) is a different point than (4, 1).
Including Spaces: Avoid spaces between the numbers and the comma or parentheses. (4, 1) is the correct format.
Not Expressing as an Ordered Pair: Simply stating x = 4 and y = 1 is not sufficient. The solution must be presented as an ordered pair (4, 1).

Reinforcing the Concept with Additional Examples

Let's consider a couple of quick examples to further reinforce the concept:

If you solve a system of equations and find x = -2 and y = 5, the solution should be expressed as (-2, 5).
If the solution is x = 0 and y = -3, the ordered pair should be written as (0, -3).

Conclusion

Expressing the solution to a system of equations as an ordered pair (a, b) with no spaces is a fundamental requirement in mathematics. It ensures clarity, facilitates graphical representation, and provides a consistent format for communication. By mastering this concept and paying attention to the details, you can confidently solve systems of equations and present your answers accurately. Remember to always verify your solution by substituting the values back into the original equations. With practice and attention to detail, you can excel in solving systems of equations and expressing the solutions correctly as ordered pairs.

The original keyword was a question asking to solve a system of equations. To make it clearer and easier to understand, we can rephrase the question as: "Solve the system of equations and express the solution as an ordered pair in the format (a, b)."