Solve Equations By Elimination: A Step-by-Step Guide
Hey everyone! Today, we're diving into solving systems of equations using the elimination method. This is a super handy technique when you have two equations with two variables, and you need to find the values of those variables that satisfy both equations simultaneously. Let's break it down with an example that was requested:
3x + 2y = 14 5x - 2y = 10
Ready? Let’s get started!
Understanding the Elimination Method
The elimination method, also known as the addition method, is all about making one of the variables disappear by adding or subtracting the equations. The key is to manipulate the equations so that the coefficients of one variable are opposites. When you add the equations, that variable gets eliminated, leaving you with a single equation in one variable. This makes it easy to solve for the remaining variable. Once you've found the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.
Before we jump into our specific problem, let's understand why this method is so powerful. Imagine you have two scales. Each scale represents an equation, and both scales are balanced. If you add equal weights to both scales, they remain balanced. Similarly, if you add the two equations together, the equality remains true. The magic happens when we strategically manipulate the equations so that adding them cancels out one of the variables. This simplifies the problem and allows us to solve for the remaining variable with ease. The beauty of the elimination method lies in its ability to transform a complex system of equations into a simple, solvable form.
Why Choose Elimination?
- Simplicity: It can be easier than substitution, especially when coefficients line up nicely.
- Efficiency: It directly eliminates a variable, streamlining the solving process.
- Versatility: Works well with linear equations and can be adapted for more complex systems.
Step-by-Step Solution
Let’s tackle the system:
3x + 2y = 14 5x - 2y = 10
Step 1: Check for Opposing Coefficients
Look at the coefficients of x and y in both equations. Notice anything special? In this case, the coefficients of y are +2 and -2. They're already opposites! This is perfect because it means we can skip a step. Sometimes, you might need to multiply one or both equations by a constant to make the coefficients of one variable opposites. But in this case, we're good to go! Recognizing these ready-made opposites can save you time and effort. It’s like finding a shortcut on a long journey.
Step 2: Add the Equations
Now, add the two equations together:
(3x + 2y) + (5x - 2y) = 14 + 10
Combine like terms:
8x = 24
The y terms cancel each other out, which is exactly what we wanted!
Step 3: Solve for x
Divide both sides by 8 to isolate x:
x = 24 / 8 x = 3
So, we've found that x = 3. Great job!
Step 4: Substitute x into One of the Original Equations
Choose either of the original equations to substitute the value of x. Let's use the first equation:
3x + 2y = 14
Substitute x = 3:
3(3) + 2y = 14 9 + 2y = 14
Step 5: Solve for y
Subtract 9 from both sides:
2y = 14 - 9 2y = 5
Divide by 2:
y = 5 / 2 y = 2.5
So, y = 2.5. Fantastic!
Step 6: Verify the Solution
To make sure we didn't make any mistakes, let's plug x = 3 and y = 2.5 into both original equations:
- Equation 1: 3x + 2y = 14
- 3(3) + 2(2.5) = 9 + 5 = 14 (Correct!)
- Equation 2: 5x - 2y = 10
- 5(3) - 2(2.5) = 15 - 5 = 10 (Correct!)
Since our solution satisfies both equations, we know we've done it right!
Another Example to Solidify Your Understanding
Let's try another example to make sure you've got the hang of it. Consider the following system:
- 4a + 3b = 17
- 2a - 3b = -5
Notice anything? The coefficients of b are +3 and -3, which are opposites! So, we can proceed directly to adding the equations:
(4a + 3b) + (2a - 3b) = 17 + (-5)
Combine like terms:
6a = 12
Divide by 6:
a = 2
Now, substitute a = 2 into the first equation:
4(2) + 3b = 17 8 + 3b = 17
Subtract 8 from both sides:
3b = 9
Divide by 3:
b = 3
So, the solution is a = 2 and b = 3. Always remember to verify your solution by plugging the values back into the original equations.
Tips and Tricks for Using Elimination
To master the elimination method, keep these tips in mind:
- Always Check for Opposites First: Before doing anything, see if any variables already have opposite coefficients. This can save you a lot of time.
- Multiply Strategically: If you don't have opposites, think about what you need to multiply each equation by to create them. Sometimes, you might need to multiply both equations.
- Keep Track of Negatives: Be careful with negative signs, especially when multiplying equations. A small mistake can throw off your entire solution.
- Stay Organized: Write neatly and keep your work organized. This will help you avoid errors and make it easier to review your steps.
- Practice Makes Perfect: The more you practice, the more comfortable you'll become with the elimination method. Try solving different systems of equations to build your skills.
When Elimination is the Best Choice
While both substitution and elimination can solve systems of equations, elimination shines in certain scenarios:
- Opposite or Easily Manipulated Coefficients: If you spot opposite coefficients or can easily create them with multiplication, elimination is usually the quicker route.
- Standard Form Equations: Elimination works well when equations are in standard form (Ax + By = C).
- Avoiding Fractions: Substitution can sometimes lead to fractions, which can be messy to work with. Elimination can help you avoid fractions in some cases.
Conclusion
And there you have it! You've successfully solved the system of equations using the elimination method. Remember, the key is to find those opposing coefficients, add the equations, solve for one variable, and then substitute back to find the other. Keep practicing, and you’ll become a pro at this in no time! This method is super useful in algebra and beyond, so mastering it will definitely pay off. Keep up the great work, and happy solving!