Solving Systems Of Equations: Find The Ordered Pair Solution

In the world of mathematics, solving systems of linear equations is a fundamental skill. It's like detective work, where you're given clues (equations) and your mission is to find the hidden values (variables) that satisfy all the clues simultaneously. This article will guide you through the process of finding the ordered pair that solves a given system of linear equations. We'll break down the problem, explore different methods, and equip you with the knowledge to tackle similar challenges.

The core concept we're dealing with is a system of linear equations. Think of it as a set of two or more equations that share the same variables. Our goal is to find the values for these variables that make all the equations true at the same time. These values, when written as a pair (x, y), represent the point where the lines represented by the equations intersect on a graph. This point of intersection is the solution to the system.

Understanding the Problem

Let's consider the specific system of equations we're tasked with solving:

1. -5x + y = 26
2. 2x - 7y = 16

Our mission is to find the ordered pair (x, y) that satisfies both of these equations. This means that when we substitute the values of x and y into each equation, the equation holds true. There are several methods we can use to solve this, including substitution, elimination, and graphing. Each method has its strengths and weaknesses, and the best choice often depends on the specific equations you're working with.

Before diving into the solution, let's consider why solving systems of equations is so important. They pop up in various real-world scenarios, from balancing chemical equations to determining the break-even point in business. Understanding how to solve them opens doors to tackling a wide range of practical problems. This is why mastering this skill is crucial for anyone pursuing STEM fields or simply wanting to enhance their problem-solving abilities. So, let's embark on this mathematical journey and uncover the solution together!

Methods to Solve Systems of Equations

There are several methods to solve systems of equations, each with its own approach. Let's delve into two common methods: substitution and elimination. Understanding these methods will equip you with the tools to solve a wide range of systems of equations.

1. The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable, which is much easier to solve. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable.

Let's apply this to our system:

1. -5x + y = 26
2. 2x - 7y = 16

Notice that the first equation is relatively easy to solve for y. We can isolate y by adding 5x to both sides:

y = 5x + 26

Now, we substitute this expression for y into the second equation:

2x - 7(5x + 26) = 16

This gives us an equation with only x as the variable. We can now simplify and solve for x:

2x - 35x - 182 = 16

-33x = 198

x = -6

With x = -6, we can substitute it back into the equation y = 5x + 26 to find y:

y = 5(-6) + 26

y = -30 + 26

y = -4

Thus, using the substitution method, we find the solution to be the ordered pair (-6, -4).

2. The Elimination Method

The elimination method involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, by adding the equations together, that variable is eliminated, leaving you with a single equation in one variable. This method is particularly useful when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant.

Let's revisit our system:

1. -5x + y = 26
2. 2x - 7y = 16

To eliminate y, we can multiply the first equation by 7:

7(-5x + y) = 7(26)

-35x + 7y = 182

Now we have the following system:

1. -35x + 7y = 182
2. 2x - 7y = 16

Adding the two equations together, the y terms cancel out:

(-35x + 7y) + (2x - 7y) = 182 + 16

-33x = 198

x = -6

As before, we substitute x = -6 into either of the original equations to solve for y. Let's use the first equation:

-5(-6) + y = 26

30 + y = 26

y = -4

Once again, we arrive at the solution (-6, -4). This demonstrates that the elimination method yields the same result as the substitution method.

Solving the System: A Step-by-Step Guide

Now, let's put our knowledge into action and solve the given system of equations step-by-step. We'll use the substitution method in this example, but remember that the elimination method could also be used.

1. The System of Equations:

We are given the following system of linear equations:

1. -5x + y = 26
2. 2x - 7y = 16

2. Isolate a Variable:

Our first step is to isolate one variable in one of the equations. Looking at the equations, it seems easiest to isolate 'y' in the first equation. To do this, we add '5x' to both sides of the equation:

-5x + y + 5x = 26 + 5x

y = 5x + 26

3. Substitute:

Now that we have 'y' isolated, we can substitute this expression for 'y' into the second equation. This means we replace 'y' in the second equation with '(5x + 26)':

2x - 7(5x + 26) = 16

4. Simplify and Solve for 'x':

Next, we simplify the equation and solve for 'x'. First, distribute the '-7' across the terms inside the parentheses:

2x - 35x - 182 = 16

Combine like terms:

-33x - 182 = 16

Add 182 to both sides:

-33x = 198

Divide both sides by -33:

x = -6

5. Substitute 'x' to Find 'y':

Now that we have the value of 'x', we can substitute it back into either of the original equations (or the equation we derived for 'y') to find 'y'. Let's use the equation 'y = 5x + 26':

y = 5(-6) + 26

y = -30 + 26

y = -4

6. Write the Solution as an Ordered Pair:

We have found that x = -6 and y = -4. Therefore, the solution to the system of equations is the ordered pair (-6, -4).

7. Verify the Solution:

To ensure our solution is correct, we can substitute the values of 'x' and 'y' back into both of the original equations. If both equations hold true, our solution is verified.

• Equation 1: -5x + y = 26

  -5(-6) + (-4) = 26

  30 - 4 = 26

  26 = 26 (True)

• Equation 2: 2x - 7y = 16

  2(-6) - 7(-4) = 16

  -12 + 28 = 16

  16 = 16 (True)

Since both equations hold true, our solution (-6, -4) is correct.

Identifying the Correct Option

Now that we've solved the system of equations, we know the solution is the ordered pair (-6, -4). Let's revisit the options provided and identify the correct one:

A. (-4, 6) B. (6, -4) C. (-4, -6) D. (-6, -4)

By comparing our solution with the options, we can clearly see that option D. (-6, -4) is the correct answer.

Conclusion

In conclusion, we've successfully navigated the process of solving a system of linear equations. We began by understanding the problem, explored different methods like substitution and elimination, and then applied the substitution method step-by-step to find the solution. The ordered pair (-6, -4) satisfies both equations in the system, making it the correct solution. Remember, the ability to solve systems of equations is a valuable skill that extends beyond the classroom, finding applications in various fields. Keep practicing, and you'll become a master problem-solver!

For further learning and practice on solving systems of equations, you can visit resources like Khan Academy's Systems of Equations.