Solving Systems Of Equations Using Elimination And Substitution

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In the realm of mathematics, solving systems of equations is a fundamental skill. These systems often represent real-world scenarios, and finding their solutions allows us to understand and predict various outcomes. This article delves into a specific example, showcasing the power of elimination and substitution methods to arrive at the solution. We'll break down each step, ensuring a clear understanding of the process.

The Power of Elimination and Substitution

When faced with a system of equations, our goal is to find the values of the variables that satisfy all equations simultaneously. Two common methods for achieving this are elimination and substitution. The elimination method involves manipulating the equations to eliminate one variable, leaving us with a single equation in one variable. This equation can then be easily solved. The substitution method, on the other hand, involves solving one equation for one variable and then substituting that expression into the other equation. This again results in a single equation in one variable.

Both methods are powerful tools, and the choice of which to use often depends on the specific structure of the equations. In some cases, elimination might be more straightforward, while in others, substitution might be the preferred approach. Understanding both methods is crucial for any aspiring mathematician or problem-solver.

Example: A System of Equations

Let's consider the following system of equations:

  1. -2x - 3y = -6
  2. y = -2

Our objective is to find the values of x and y that satisfy both equations. Notice that the second equation already gives us the value of y. This makes the substitution method particularly appealing in this case.

Step 1: Substitution

The second equation tells us that y = -2. We can substitute this value into the first equation:

-2x - 3(-2) = -6

This substitution eliminates the variable y from the first equation, leaving us with an equation solely in terms of x.

Step 2: Simplify and Solve for x

Now, let's simplify the equation:

-2x + 6 = -6

Subtract 6 from both sides:

-2x = -12

Finally, divide both sides by -2:

x = 6

We have now found the value of x.

Step 3: The Solution

We have determined that x = 6 and y = -2. Therefore, the solution to the system of equations is the ordered pair (6, -2). This ordered pair represents the point where the two lines represented by the equations intersect on a graph.

Solving for x After Substitution: A Detailed Walkthrough

The given problem focuses on a specific part of the solution process: solving for x after substituting the value of y. Let's revisit the relevant steps.

We started with the equation:

-2x - 3y = -6

And we were given that y = -2. Substituting this value, we get:

-2x - 3(-2) = -6

This is where the problem picks up. The next step is to simplify the equation. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we need to perform the multiplication before we can proceed:

-2x + 6 = -6

Now, we want to isolate the term with x. To do this, we subtract 6 from both sides of the equation. This maintains the equality and moves us closer to solving for x:

-2x + 6 - 6 = -6 - 6

-2x = -12

Finally, to solve for x, we divide both sides of the equation by -2. This is the coefficient of x, and dividing by it will give us the value of x:

-2x / -2 = -12 / -2

x = 6

Therefore, the solution for x is 6.

The Solution to the System

Since we already know that y = -2, the complete solution to the system of equations is x = 6 and y = -2. This can be written as the ordered pair (6, -2). This ordered pair represents the point where the two lines represented by the equations intersect on a graph. To further solidify our understanding, we can verify this solution by substituting these values back into the original equations.

Verifying the Solution

Verification is a crucial step in solving systems of equations. It ensures that our solution is correct and that we haven't made any errors in our calculations. To verify, we substitute the values of x and y back into the original equations and check if the equations hold true.

Let's take our solution (6, -2) and substitute it into the first equation:

-2x - 3y = -6

-2(6) - 3(-2) = -6

-12 + 6 = -6

-6 = -6

The equation holds true. Now, let's substitute the values into the second equation:

y = -2

-2 = -2

This equation also holds true. Since our solution satisfies both equations, we can confidently say that (6, -2) is the correct solution to the system of equations.

Key Concepts and Takeaways

This example highlights several key concepts in solving systems of equations:

  • Substitution: Recognizing when substitution is a convenient method is crucial. If one equation is already solved for a variable, substitution is often the easiest approach.
  • Order of Operations: Following the correct order of operations (PEMDAS/BODMAS) is essential for accurate calculations.
  • Isolating Variables: The goal in solving for a variable is to isolate it on one side of the equation. This involves using inverse operations (addition/subtraction, multiplication/division) to undo the operations performed on the variable.
  • Verification: Always verify your solution by substituting the values back into the original equations. This helps catch errors and ensures accuracy.

Understanding these concepts and practicing with various examples will solidify your skills in solving systems of equations. Remember, mathematics is a skill that improves with practice, so keep exploring and challenging yourself!

Conclusion

Solving systems of equations is a fundamental skill in mathematics, with applications in various fields. This article has demonstrated how to solve a system of equations using a combination of substitution and algebraic manipulation. By understanding the steps involved and practicing regularly, you can master this skill and confidently tackle more complex problems. The key takeaway is to break down the problem into smaller, manageable steps and to always verify your solution to ensure accuracy. Keep exploring the world of mathematics, and you'll discover its beauty and power in solving real-world problems.