Solve The System Of Equations: 2x + 7y = -7, -4x - 3y = -19. Express The Answer As An Ordered Pair (a, B).

Jun 21, 2025 by ADMIN 107 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. This article will guide you through the process of solving a system of two linear equations with two variables. We'll break down each step, ensuring you understand the logic and techniques involved. Our goal is to find the values of the variables that satisfy both equations simultaneously, expressing the solution as an ordered pair in the format (a, b). Let's dive in!

Understanding Systems of Equations

Before we tackle the problem, it's crucial to grasp what a system of equations represents. A system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the set of values for the variables that make all the equations true. In our case, we have two linear equations:

2x + 7y = -7
-4x - 3y = -19

Our mission is to find the values of x and y that satisfy both of these equations. There are several methods to solve systems of equations, including substitution, elimination, and graphing. We'll focus on the elimination method in this guide, as it's often an efficient approach for this type of problem.

The Elimination Method: A Detailed Walkthrough

The elimination method involves manipulating the equations so that when they are added together, one of the variables is eliminated. This leaves us with a single equation with one variable, which we can easily solve. Let's apply this to our system:

Step 1: Choose a Variable to Eliminate

In our system, we can choose to eliminate either x or y. Looking at the coefficients, it seems easier to eliminate x because the coefficients of x in the two equations are 2 and -4. Notice that -4 is a multiple of 2. This suggests we can multiply the first equation by 2 to make the x coefficients opposites.

Step 2: Multiply One or Both Equations

We'll multiply the first equation (2x + 7y = -7) by 2:

2 * (2x + 7y) = 2 * (-7)
4x + 14y = -14

Now our system looks like this:

4x + 14y = -14
-4x - 3y = -19

Notice that the coefficients of x are now 4 and -4, which are opposites.

Step 3: Add the Equations

Now we add the two equations together:

(4x + 14y) + (-4x - 3y) = -14 + (-19)

This simplifies to:

11y = -33

The x terms have been eliminated, as planned!

Step 4: Solve for the Remaining Variable

We now have a simple equation with one variable, y. To solve for y, we divide both sides by 11:

11y / 11 = -33 / 11
y = -3

So, we've found that y = -3.

Step 5: Substitute to Find the Other Variable

Now that we know the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first original equation (2x + 7y = -7):

2x + 7(-3) = -7
2x - 21 = -7

Add 21 to both sides:

2x = 14

Divide by 2:

x = 7

So, we've found that x = 7.

Step 6: Write the Solution as an Ordered Pair

Finally, we express our solution as an ordered pair in the format (a, b), where a is the value of x and b is the value of y. In our case, x = 7 and y = -3, so the solution is (7, -3).

Verification: Ensuring the Solution is Correct

It's always a good practice to verify your solution by plugging the values of x and y back into both original equations to make sure they hold true.

Equation 1: 2x + 7y = -7

2(7) + 7(-3) = -7
14 - 21 = -7
-7 = -7  (True)

Equation 2: -4x - 3y = -19

-4(7) - 3(-3) = -19
-28 + 9 = -19
-19 = -19  (True)

Since the values x = 7 and y = -3 satisfy both equations, our solution is correct.

Alternative Methods: Exploring Substitution

While we focused on the elimination method, the substitution method is another powerful technique for solving systems of equations. This method involves solving one equation for one variable and then substituting that expression into the other equation. Let's briefly illustrate how it could be applied to our system:

Solve for one variable: Choose one equation and solve it for one of the variables. For instance, let's solve the first equation (2x + 7y = -7) for x:
```
2x = -7 - 7y
x = (-7 - 7y) / 2
```
Substitute: Substitute this expression for x into the second equation (-4x - 3y = -19):
```
-4((-7 - 7y) / 2) - 3y = -19
```
Solve for y: Simplify and solve the equation for y:
```
14 + 14y - 3y = -19
11y = -33
y = -3
```
Substitute back: Substitute the value of y back into the expression for x:
```
x = (-7 - 7(-3)) / 2
x = ( -7 + 21) / 2
x = 7
```

As you can see, the substitution method leads to the same solution (7, -3).

Common Mistakes to Avoid

Solving systems of equations can be tricky, and it's easy to make mistakes. Here are some common errors to watch out for:

Sign Errors: Pay close attention to the signs of the coefficients and constants. A simple sign error can throw off the entire solution.
Arithmetic Errors: Double-check your arithmetic calculations, especially when multiplying, adding, or dividing equations.
Incorrect Substitution: When using the substitution method, make sure you substitute the expression into the correct equation and simplify carefully.
Forgetting to Distribute: When multiplying an equation by a constant, remember to distribute the constant to all terms in the equation.

By being mindful of these potential pitfalls, you can increase your accuracy and confidence in solving systems of equations.

Real-World Applications: Where Systems of Equations Matter

Systems of equations aren't just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

Economics: Supply and demand models often involve systems of equations to determine equilibrium prices and quantities.
Engineering: Circuit analysis, structural design, and fluid dynamics problems frequently require solving systems of equations.
Computer Graphics: Systems of equations are used in 3D modeling, animation, and rendering.
Chemistry: Balancing chemical equations involves solving a system of equations.
Navigation: GPS systems use systems of equations to determine a user's location.

Understanding systems of equations is therefore a valuable skill in many fields.

Conclusion: Mastering the Art of Solving Systems of Equations

In this comprehensive guide, we've explored the process of solving a system of two linear equations using the elimination method. We've broken down each step, from choosing a variable to eliminate to verifying the solution. We've also touched upon the substitution method and highlighted common mistakes to avoid. Remember, practice is key to mastering this skill. By working through various examples, you'll develop a strong intuition for solving systems of equations and gain confidence in your abilities. So, keep practicing, and you'll become a pro at finding those ordered pair solutions!

Answer: (7,-3)