Solving 2x + 3y = 3 And 3x - Y = 10 By Substitution Method A Comprehensive Guide
Solving systems of equations is a fundamental concept in algebra, and the substitution method is a powerful tool for tackling these problems. In this comprehensive guide, we'll walk through the process of solving the system of equations 2x + 3y = 3 and 3x - y = 10 using the substitution method. So, guys, let's dive in and make math a little less intimidating!
Understanding Systems of Equations
Before we jump into the substitution method, let's make sure we're all on the same page about what a system of equations is. A system of equations is simply a set of two or more equations that share the same variables. The goal is to find values for these variables that satisfy all equations in the system simultaneously. Think of it like finding the perfect meeting point for multiple lines on a graph. In our case, we have two equations:
- 2x + 3y = 3
- 3x - y = 10
We need to find the values of 'x' and 'y' that make both of these equations true at the same time. There are several methods to solve systems of equations, including graphing, elimination, and, of course, substitution. Each method has its strengths and weaknesses, but the substitution method is particularly useful when one equation can be easily solved for one variable in terms of the other. This brings us to the heart of our topic – the substitution method.
The Substitution Method A Step-by-Step Approach
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation that we can easily solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. It's like a clever puzzle where we rearrange pieces to reveal the solution. Let's break down the steps with our example:
Step 1: Solve one equation for one variable
The first step is to choose one of the equations and solve it for one of the variables. Look for an equation where a variable has a coefficient of 1 or -1, as this will make the algebra simpler. In our system:
- 2x + 3y = 3
- 3x - y = 10
The second equation, 3x - y = 10, looks like a good candidate because the 'y' term has a coefficient of -1. Let's solve this equation for 'y'. To isolate 'y', we'll first subtract 3x from both sides:
3x - y - 3x = 10 - 3x
-y = 10 - 3x
Now, to get 'y' by itself, we'll multiply both sides by -1:
(-1)(-y) = (-1)(10 - 3x)
y = -10 + 3x
We can also write this as:
y = 3x - 10
So, we've successfully solved the second equation for 'y'. This expression for 'y' is the key that unlocks the next step.
Step 2: Substitute the expression into the other equation
Now that we have an expression for 'y' in terms of 'x', we'll substitute this expression into the other equation (the one we didn't use in Step 1). This is where the magic of the substitution method happens. We're replacing 'y' in the first equation with its equivalent expression in terms of 'x'.
Our first equation is:
2x + 3y = 3
We'll substitute 'y' with '3x - 10':
2x + 3(3x - 10) = 3
Notice that we now have an equation with only one variable, 'x'. This is a crucial step because we can now solve for 'x'. This substitution effectively transforms our system of two equations into a single equation that we can handle. The next part is all about simplifying and isolating 'x'.
Step 3: Solve the resulting equation
We now have the equation:
2x + 3(3x - 10) = 3
Let's simplify and solve for 'x'. First, we'll distribute the 3:
2x + 9x - 30 = 3
Next, combine like terms:
11x - 30 = 3
Now, add 30 to both sides:
11x = 33
Finally, divide both sides by 11:
x = 3
Great! We've found the value of 'x'. This is one half of our solution. Now we just need to find the value of 'y'. With 'x' in hand, the next step is relatively straightforward.
Step 4: Substitute the value back to find the other variable
We've found that x = 3. Now we need to find the value of 'y'. We can substitute this value of 'x' into either of the original equations or the expression we found for 'y' in Step 1. The easiest option is usually the expression we found in Step 1:
y = 3x - 10
Substitute x = 3:
y = 3(3) - 10
y = 9 - 10
y = -1
So, we've found that y = -1. We now have both values, x = 3 and y = -1. It's always a good idea to check our solution to make sure it satisfies both original equations.
Step 5: Check your solution
To check our solution, we'll substitute x = 3 and y = -1 into both of the original equations:
Equation 1: 2x + 3y = 3
2(3) + 3(-1) = 6 - 3 = 3 (Correct!)
Equation 2: 3x - y = 10
3(3) - (-1) = 9 + 1 = 10 (Correct!)
Since our values satisfy both equations, we can confidently say that our solution is correct. We've successfully navigated the substitution method and found the solution to our system of equations.
The Solution
The solution to the system of equations 2x + 3y = 3 and 3x - y = 10 is x = 3 and y = -1. We can write this as an ordered pair (3, -1). This point represents the intersection of the two lines represented by the equations on a graph. The substitution method has allowed us to find this point algebraically, without needing to draw a graph. Remember, this solution is the only pair of values for 'x' and 'y' that will make both equations true simultaneously.
When to Use the Substitution Method
The substitution method is particularly useful when one of the equations can be easily solved for one variable. This often happens when a variable has a coefficient of 1 or -1, as in our example. However, the substitution method can be used for any system of equations. If neither equation is easily solved for a variable, the elimination method might be a better choice. The best method often depends on the specific equations in the system. The key is to choose the method that minimizes the amount of algebra involved and reduces the chances of making a mistake.
Advantages and Disadvantages of the Substitution Method
Like any problem-solving tool, the substitution method has its pros and cons. Understanding these can help you decide when to use it and when to opt for another approach.
Advantages
- Effective for simple equations: It's very efficient when one equation is already solved or can be easily solved for a variable.
- Clear and straightforward: The steps are logical and easy to follow once you understand the concept.
- Reduces variables: It systematically reduces a two-variable problem into a single-variable problem.
Disadvantages
- Can be cumbersome for complex equations: If the equations involve fractions or complex expressions, the substitution method can become quite messy.
- May lead to errors: The algebraic manipulations involved can be prone to errors if not done carefully.
- Not always the most efficient: For some systems, the elimination method might be faster and less error-prone.
Tips and Tricks for Mastering the Substitution Method
Here are a few tips and tricks to help you master the substitution method and avoid common mistakes:
- Choose wisely: When deciding which variable to solve for, look for the one with the simplest coefficient (1 or -1). This will minimize fractions and simplify the algebra.
- Substitute carefully: Make sure you substitute the expression into the other equation, not the one you used to solve for the variable. Substituting back into the same equation will lead to a trivial identity (like 0 = 0) and won't help you find the solution.
- Distribute properly: When substituting an expression into an equation, remember to distribute any coefficients correctly.
- Check your work: Always check your solution by substituting the values back into both original equations. This will catch any arithmetic errors and ensure that your solution is correct.
- Practice, practice, practice: The more you practice, the more comfortable you'll become with the substitution method. Work through a variety of examples to build your skills and confidence.
Common Mistakes to Avoid
Even with a solid understanding of the steps, it's easy to make mistakes when using the substitution method. Here are a few common pitfalls to watch out for:
- Substituting into the same equation: As mentioned earlier, substituting back into the same equation you used to solve for a variable is a common mistake. This will not give you any new information.
- Forgetting to distribute: When substituting an expression into an equation, make sure you distribute any coefficients properly. For example, if you have 2(3x - 1), you need to distribute the 2 to both terms inside the parentheses.
- Arithmetic errors: Simple arithmetic errors are easy to make, especially when dealing with negative numbers. Double-check your calculations to avoid these mistakes.
- Not checking your solution: Always check your solution by substituting the values back into both original equations. This is the best way to catch any errors.
Real-World Applications of Systems of Equations
Systems of equations aren't just abstract mathematical concepts; they have many real-world applications. Here are a few examples:
- Economics: Supply and demand curves can be modeled as systems of equations. The solution to the system represents the equilibrium price and quantity.
- Engineering: Electrical circuits can be analyzed using systems of equations. The solution to the system gives the currents and voltages in the circuit.
- Physics: Many physics problems, such as those involving motion and forces, can be solved using systems of equations.
- Chemistry: Balancing chemical equations often involves solving systems of equations.
- Everyday life: Systems of equations can be used to solve problems like mixing solutions, calculating costs, and determining distances.
Conclusion
The substitution method is a valuable tool for solving systems of equations. By following the steps outlined in this guide and practicing regularly, you can master this technique and confidently tackle a wide range of problems. Remember to choose your variables wisely, substitute carefully, and always check your solution. With a little practice, you'll be solving systems of equations like a pro! So go forth, guys, and conquer those equations!
