Is (2,2) A Solution To The System Of Equations?
Hey guys! Let's dive into a math problem today where we need to figure out if a specific point is a solution to a system of equations. Specifically, we want to know if the point (2, 2) is a solution to the following system:
y = -4x + 6
3x + 4y = -2
To determine if (2, 2) is a solution, we need to substitute x = 2 and y = 2 into both equations. If both equations are true after the substitution, then (2, 2) is indeed a solution. If at least one equation is false, then (2, 2) is not a solution. Let's break it down step-by-step.
Step 1: Substitute into the First Equation
The first equation is y = -4x + 6. We substitute x = 2 and y = 2 into this equation:
2 = -4(2) + 6
Now, we simplify the right side of the equation:
2 = -8 + 6
2 = -2
Since 2 is not equal to -2, the first equation is false when x = 2 and y = 2. This means that the point (2, 2) does not satisfy the first equation. Therefore, we don't even need to check the second equation to determine that (2, 2) is not a solution to the system. However, for the sake of completeness and understanding, let's proceed with checking the second equation as well.
Step 2: Substitute into the Second Equation
The second equation is 3x + 4y = -2. Again, we substitute x = 2 and y = 2 into this equation:
3(2) + 4(2) = -2
Now, we simplify the left side of the equation:
6 + 8 = -2
14 = -2
Since 14 is not equal to -2, the second equation is also false when x = 2 and y = 2. This confirms that the point (2, 2) does not satisfy the second equation either. Since (2, 2) does not satisfy either equation in the system, it is definitively not a solution to the system of equations.
Conclusion
In conclusion, by substituting x = 2 and y = 2 into both equations, we found that neither equation holds true. Therefore, (2, 2) is not a solution to the system of equations:
y = -4x + 6
3x + 4y = -2
So the final answer is No.
Why This Matters
Understanding how to verify solutions to systems of equations is a fundamental concept in algebra. It allows us to check if a given point satisfies all the equations in the system, which is crucial in many mathematical and real-world applications. For example, in economics, you might have a system of equations representing supply and demand, and you want to find the equilibrium point (the point where supply equals demand). Verifying solutions helps ensure that your equilibrium point is correct.
The Importance of Accuracy
When solving systems of equations, it’s super important to be accurate with your calculations. A small mistake can lead to an incorrect solution, which can have significant consequences depending on the application. Always double-check your work and make sure you're substituting values correctly. Using tools like calculators or software can help reduce errors, but understanding the underlying concepts is key. Remember, the point must satisfy all equations in the system to be a valid solution. If it fails even one, it's not a solution!
Alternative Methods for Solving Systems of Equations
While we focused on verifying a potential solution, let's briefly touch on some other methods for actually solving systems of equations. Knowing these methods can help you find solutions when they're not given to you.
1. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which you can solve. Once you find the value of that variable, you can substitute it back into either of the original equations to find the value of the other variable. For example, using the given system:
y = -4x + 6
3x + 4y = -2
We already have the first equation solved for y. We can substitute -4x + 6 for y in the second equation:
3x + 4(-4x + 6) = -2
3x - 16x + 24 = -2
-13x = -26
x = 2
Now, substitute x = 2 back into the first equation:
y = -4(2) + 6
y = -8 + 6
y = -2
So the solution is (2, -2).
2. Elimination Method
The elimination method involves adding or subtracting the equations to eliminate one of the variables. This often requires multiplying one or both equations by a constant to make the coefficients of one variable match. Let’s use the same system again:
y = -4x + 6
3x + 4y = -2
First, rewrite the first equation to align the variables:
4x + y = 6
3x + 4y = -2
Multiply the first equation by -4 to eliminate y:
-16x - 4y = -24
3x + 4y = -2
Add the two equations:
-13x = -26
x = 2
Substitute x = 2 back into the equation 4x + y = 6:
4(2) + y = 6
8 + y = 6
y = -2
Again, the solution is (2, -2).
3. Graphical Method
The graphical method involves graphing both equations on the same coordinate plane. The point where the two lines intersect is the solution to the system. This method is particularly useful for visualizing the solution, but it may not be as accurate as the algebraic methods, especially if the solution involves fractions or decimals. If you graph y = -4x + 6 and 3x + 4y = -2, you'll see they intersect at (2, -2).
Practice Makes Perfect
The best way to get comfortable with systems of equations is to practice! Try solving different systems using all three methods—substitution, elimination, and graphing. Pay attention to the details, double-check your work, and don't be afraid to ask for help when you get stuck. Remember, math is like learning a new language, and the more you practice, the more fluent you'll become. Keep at it, and you'll master these concepts in no time!
So, next time someone asks you if a point is a solution to a system of equations, you'll be ready to tackle it with confidence. Keep practicing, and you'll become a system-solving superstar! Remember the key steps: substitute, simplify, and check if the equations hold true. You got this!