Solving 3x+4y=20 And 4x-3y=10 Elimination And Substitution Methods
Hey guys! Ever found yourselves staring at a system of equations and feeling totally lost? Don't worry, we've all been there. Today, we're going to break down a common type of problem in mathematics: solving a system of equations. Specifically, we'll tackle the system 3x + 4y = 20 and 4x - 3y = 10 using two powerful methods: elimination and substitution. So, grab your pencils and let's dive in!
Understanding Systems of Equations
Before we jump into the solutions, let's quickly recap what a system of equations actually is. At its core, a system of equations is just a set of two or more equations that share the same variables. In our case, we have two equations with two variables, x and y. The goal is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding the point where two lines intersect on a graph – that point represents the solution that works for both equations.
Systems of equations pop up everywhere in the real world. From calculating the break-even point for a business to determining the trajectory of a rocket, these systems help us model and solve a wide range of problems. Mastering the techniques to solve them is a crucial skill in mathematics and beyond.
Method 1: The Elimination Method
The elimination method is all about, well, eliminating one of the variables! The key idea is to manipulate the equations so that the coefficients of either x or y are opposites. This way, when we add the equations together, one variable cancels out, leaving us with a single equation in one variable. Let's see how it works with our system:
3x + 4y = 20
4x - 3y = 10
Our goal is to make either the x coefficients or the y coefficients opposites. Let's focus on the y coefficients. We have 4y in the first equation and -3y in the second. To make them opposites, we can multiply the first equation by 3 and the second equation by 4. This will give us 12y and -12y, which are perfect opposites.
Multiplying the first equation by 3, we get:
(3 * (3x + 4y) = 3 * 20) which simplifies to 9x + 12y = 60
Multiplying the second equation by 4, we get:
(4 * (4x - 3y) = 4 * 10) which simplifies to 16x - 12y = 40
Now we have a new system of equations:
9x + 12y = 60
16x - 12y = 40
See how the y coefficients are opposites? Now we can add the two equations together. When we add the left-hand sides, the 12y and -12y terms cancel out. When we add the right-hand sides, we get 60 + 40 = 100. So, adding the equations gives us:
(9x + 12y) + (16x - 12y) = 60 + 40 which simplifies to 25x = 100
Now we have a simple equation with just one variable! To solve for x, we divide both sides by 25:
25x / 25 = 100 / 25 which simplifies to x = 4
Great! We've found the value of x. Now we need to find the value of y. We can do this by substituting the value of x (which is 4) into either of our original equations. Let's use the first equation, 3x + 4y = 20:
Substituting x = 4, we get:
3 * 4 + 4y = 20 which simplifies to 12 + 4y = 20
Now we solve for y. Subtract 12 from both sides:
12 + 4y - 12 = 20 - 12 which simplifies to 4y = 8
Divide both sides by 4:
4y / 4 = 8 / 4 which simplifies to y = 2
And there we have it! We've found the values of x and y using the elimination method. The solution to the system of equations is x = 4 and y = 2. We can write this as an ordered pair (4, 2).
Key Steps in the Elimination Method:
- Identify the variable you want to eliminate.
- Multiply one or both equations by constants so that the coefficients of the variable you want to eliminate are opposites.
- Add the equations together. This should eliminate one variable.
- Solve the resulting equation for the remaining variable.
- Substitute the value you found back into one of the original equations to solve for the other variable.
- Check your solution by substituting both values into both original equations.
Method 2: The Substitution Method
The substitution method takes a slightly different approach. Instead of eliminating a variable, we solve one equation for one variable in terms of the other and then substitute that expression into the other equation. This leaves us with a single equation in one variable, which we can then solve.
Let's use the same system of equations:
3x + 4y = 20
4x - 3y = 10
First, we need to choose one equation and solve for one variable. Let's choose the first equation, 3x + 4y = 20, and solve for x. To do this, we'll isolate x on one side of the equation.
Subtract 4y from both sides:
3x + 4y - 4y = 20 - 4y which simplifies to 3x = 20 - 4y
Divide both sides by 3:
3x / 3 = (20 - 4y) / 3 which simplifies to x = (20 - 4y) / 3
Now we have an expression for x in terms of y. This is the crucial step in the substitution method. We're going to substitute this expression for x into the other equation, the second equation 4x - 3y = 10.
Substituting x = (20 - 4y) / 3, we get:
4 * ((20 - 4y) / 3) - 3y = 10
This looks a bit messy, but don't worry! We can simplify it. First, let's distribute the 4:
(80 - 16y) / 3 - 3y = 10
To get rid of the fraction, we can multiply both sides of the equation by 3:
3 * ((80 - 16y) / 3 - 3y) = 3 * 10 which simplifies to 80 - 16y - 9y = 30
Now we have a simpler equation in terms of y. Combine the y terms:
80 - 25y = 30
Subtract 80 from both sides:
80 - 25y - 80 = 30 - 80 which simplifies to -25y = -50
Divide both sides by -25:
-25y / -25 = -50 / -25 which simplifies to y = 2
Hey, look at that! We found y = 2, which is the same value we got using the elimination method. That's a good sign! Now we need to find the value of x. We can substitute the value of y (which is 2) back into our expression for x: x = (20 - 4y) / 3
Substituting y = 2, we get:
x = (20 - 4 * 2) / 3 which simplifies to x = (20 - 8) / 3 which simplifies to x = 12 / 3 which simplifies to x = 4
Woohoo! We've found x = 4, which again matches our result from the elimination method. So, using the substitution method, we also get the solution x = 4 and y = 2, or the ordered pair (4, 2).
Key Steps in the Substitution Method:
- Choose one equation and solve for one variable in terms of the other.
- Substitute the expression you found into the other equation.
- Solve the resulting equation for the remaining variable.
- Substitute the value you found back into your expression from step 1 to solve for the other variable.
- Check your solution by substituting both values into both original equations.
Comparing Elimination and Substitution
So, we've solved the same system of equations using two different methods. Which method is better? Well, it depends! Both elimination and substitution have their strengths and weaknesses.
- Elimination is often a good choice when the coefficients of one of the variables are already opposites or are easy to make opposites. It can be a more straightforward method when dealing with equations where the variables are neatly aligned.
- Substitution is often a good choice when one of the equations is already solved for one variable or can be easily solved for one variable. It's also useful when dealing with more complex systems of equations where elimination might be more cumbersome.
In the end, the best method is the one that you feel most comfortable with and that seems most efficient for the specific problem you're facing. The more you practice, the better you'll become at recognizing which method is the best fit for each situation.
Checking Our Solution
No matter which method you use, it's always a good idea to check your solution! This helps ensure that you haven't made any mistakes along the way. To check our solution, we'll substitute the values we found, x = 4 and y = 2, into both of our original equations.
Let's start with the first equation, 3x + 4y = 20:
Substituting x = 4 and y = 2, we get:
3 * 4 + 4 * 2 = 20 which simplifies to 12 + 8 = 20 which simplifies to 20 = 20
Great! The equation holds true. Now let's check the second equation, 4x - 3y = 10:
Substituting x = 4 and y = 2, we get:
4 * 4 - 3 * 2 = 10 which simplifies to 16 - 6 = 10 which simplifies to 10 = 10
Excellent! The second equation also holds true. Since our solution satisfies both original equations, we can be confident that it's correct.
Conclusion
Solving systems of equations might seem daunting at first, but with practice, you'll become a pro! We've explored two powerful methods – elimination and substitution – for solving the system 3x + 4y = 20 and 4x - 3y = 10. We found that both methods lead to the same solution: x = 4 and y = 2, or the ordered pair (4, 2). Remember to check your solutions to avoid errors, guys!
The key takeaway here is that understanding different problem-solving techniques gives you flexibility and confidence. So, keep practicing, and you'll be conquering systems of equations in no time! Happy solving!
