Solving Systems Of Equations: A Step-by-Step Guide

by Alex Johnson 51 views

Have you ever encountered a problem where you have two equations with two unknowns, and you need to find the values that satisfy both equations simultaneously? This is a classic scenario in algebra known as solving a system of equations. In this guide, we'll walk you through the process of solving such systems, using the specific example:

y = 2x - 3.5
x - 2y = -14

We'll break down the steps, explain the concepts, and make sure you're comfortable tackling these types of problems. So, let's dive in!

Understanding Systems of Equations

First, let's clarify what a system of equations actually is. A system of equations is a set of two or more equations containing 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 simpler terms, it's the point where the lines represented by the equations intersect on a graph.

There are several methods to solve systems of equations, but we'll focus on the substitution method here, as it's particularly effective for the given problem.

The Substitution Method: A Detailed Walkthrough

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, allowing you to solve for the remaining one. Let's apply this method to our system:

y = 2x - 3.5  (Equation 1)
x - 2y = -14 (Equation 2)

Step 1: Solve one equation for one variable

Notice that Equation 1 is already solved for y. This makes our job easier! We have:

y = 2x - 3.5

Step 2: Substitute the expression into the other equation

Now, we'll substitute the expression 2x - 3.5 for y in Equation 2:

x - 2(2x - 3.5) = -14

This single equation now only contains the variable x, which we can solve.

Step 3: Solve the resulting equation

Let's simplify and solve the equation for x:

x - 4x + 7 = -14
-3x + 7 = -14
-3x = -21
x = 7

So, we've found that x = 7.

Step 4: Substitute the value back to find the other variable

Now that we know x = 7, we can substitute this value back into either Equation 1 or Equation 2 to find y. Let's use Equation 1, as it's already solved for y:

y = 2(7) - 3.5
y = 14 - 3.5
y = 10.5

Therefore, y = 10.5.

Step 5: Check the solution

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

Let's check Equation 1:

10.  5 = 2(7) - 3.5
11.  5 = 14 - 3.5
12.  5 = 10.5  (True)

And now Equation 2:

7 - 2(10.5) = -14
7 - 21 = -14
-14 = -14 (True)

Since our solution satisfies both equations, we've confirmed that it's correct.

The Solution: (7, 10.5)

We've successfully solved the system of equations! The solution is x = 7 and y = 10.5. This can be written as an ordered pair: (7, 10.5).

Looking back at the options provided:

A. (-7, 3.5) B. (3.5, -7) C. (7, 10.5) D. (10.5, 7)

We can see that option C, (7, 10.5), is the correct answer.

Alternative Methods for Solving Systems of Equations

While we focused on the substitution method in this example, it's important to be aware of other methods for solving systems of equations. Here are a couple of common alternatives:

1. Elimination Method

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. Then, you add the equations together, which eliminates that variable. You can then solve for the remaining variable and substitute back to find the other.

This method is particularly useful when the equations are in standard form (Ax + By = C).

2. Graphing

The graphing method involves graphing both equations on the same coordinate plane. The point where the lines intersect represents the solution to the system. This method is visually intuitive but may not be the most accurate for non-integer solutions.

Tips and Tricks for Solving Systems of Equations

Solving systems of equations can sometimes be tricky, but here are a few tips to keep in mind:

  • Check your work: Always double-check your solution by substituting the values back into the original equations.
  • Choose the easiest method: Consider the form of the equations and choose the method that seems most efficient. Substitution is good when one equation is already solved for a variable, while elimination is good when the equations are in standard form.
  • Be careful with signs: Pay close attention to the signs (positive and negative) when manipulating equations.
  • Stay organized: Keep your work neat and organized to avoid errors.

Real-World Applications of Systems of Equations

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

  • Economics: Determining supply and demand equilibrium.
  • Engineering: Designing structures and circuits.
  • Physics: Solving problems involving motion and forces.
  • Chemistry: Balancing chemical equations.
  • Computer Graphics: Creating 3D models and animations.

Practice Makes Perfect

The best way to master solving systems of equations is to practice! Work through various examples, try different methods, and don't be afraid to make mistakes. Each mistake is a learning opportunity.

Conclusion

Solving systems of equations is a fundamental skill in algebra with wide-ranging applications. We've covered the substitution method in detail, explored alternative methods, and provided tips for success. Remember, the key is to understand the underlying concepts and practice consistently. With a little effort, you'll become a pro at solving systems of equations!

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