Is (3, 6) A Solution To The System Of Equations?
In mathematics, determining whether a given point is a solution to a system of equations is a fundamental concept in algebra. This involves substituting the coordinates of the point into each equation in the system and verifying if the equations hold true. In this article, we will explore how to determine if the point (3, 6) is a solution to the following system of equations:
y = (4/3)x + 2
y = 3x + 3
This process is crucial for understanding the graphical representation of solutions and their significance in various mathematical and real-world applications. Let's dive into the steps involved in verifying the solution.
Understanding Systems of Equations and Solutions
Before we delve into the specifics of the given problem, it's essential to understand what a system of equations is and what constitutes a solution. 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 satisfy all equations in the system simultaneously. Graphically, the solution represents the point(s) where the lines or curves described by the equations intersect.
In our case, we have a system of two linear equations:
y = (4/3)x + 2
y = 3x + 3
Each equation represents a straight line on the coordinate plane. A solution to this system would be a point (x, y) that lies on both lines. To determine if (3, 6) is a solution, we need to substitute x = 3 and y = 6 into both equations and check if the equations hold true.
The Significance of Solutions
Solutions to systems of equations are not just abstract mathematical concepts; they have significant applications in various fields. In economics, they can represent market equilibrium points where supply equals demand. In engineering, they can help determine the optimal design parameters for structures and systems. In computer science, they are used in algorithms for optimization and problem-solving. Understanding how to find and verify solutions is therefore a crucial skill in many disciplines.
Methods for Solving Systems of Equations
There are several methods for solving systems of equations, including:
- Substitution: Solve one equation for one variable and substitute that expression into the other equation.
- Elimination: Add or subtract the equations to eliminate one variable.
- Graphing: Plot the equations on a coordinate plane and find the points of intersection.
- Matrix Methods: Use matrix operations to solve systems of linear equations.
In this article, we are focusing on verifying a given solution rather than solving the system from scratch. However, it's important to be aware of these different methods for solving systems of equations.
Step-by-Step Verification Process
Now, let's proceed with the step-by-step verification process to determine if (3, 6) is indeed a solution to our system of equations. The key is to substitute the x and y values into each equation separately and check if the equality holds.
Step 1: Substitute into the First Equation
The first equation is:
y = (4/3)x + 2
Substitute x = 3 and y = 6 into the equation:
6 = (4/3)(3) + 2
Now, simplify the right-hand side:
6 = 4 + 2
6 = 6
The equation holds true. This means that the point (3, 6) lies on the line represented by the first equation.
Step 2: Substitute into the Second Equation
The second equation is:
y = 3x + 3
Substitute x = 3 and y = 6 into the equation:
6 = 3(3) + 3
Now, simplify the right-hand side:
6 = 9 + 3
6 = 12
In this case, the equation does not hold true. 6 is not equal to 12. This indicates that the point (3, 6) does not lie on the line represented by the second equation.
Analyzing the Results
From our verification process, we found that the point (3, 6) satisfies the first equation but not the second equation. For a point to be a solution to the system of equations, it must satisfy all equations in the system. Since (3, 6) does not satisfy both equations, we can conclude that it is not a solution to the system.
Common Mistakes and How to Avoid Them
When verifying solutions to systems of equations, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results.
Mistake 1: Only Checking One Equation
One of the most common errors is to substitute the point into only one equation and conclude that it is a solution if that equation holds true. As we've seen, a solution must satisfy all equations in the system. Always remember to check the point in every equation.
Mistake 2: Arithmetic Errors
Simple arithmetic errors during the substitution and simplification process can lead to incorrect conclusions. Double-check your calculations, especially when dealing with fractions or negative numbers. It's a good practice to write down each step clearly to minimize the chances of making a mistake.
Mistake 3: Misinterpreting the Question
Ensure you understand exactly what the question is asking. Are you being asked to solve the system, verify a solution, or perform some other operation? Misinterpreting the question can lead you down the wrong path.
Mistake 4: Forgetting the Order of Operations
When simplifying expressions, remember to follow the order of operations (PEMDAS/BODMAS). Perform operations in the correct order to avoid errors.
Tips for Accuracy
To avoid these mistakes, consider the following tips:
- Write down each step clearly: This helps you keep track of your work and makes it easier to spot errors.
- Double-check your calculations: After simplifying an expression, take a moment to review your work and ensure you haven't made any arithmetic errors.
- Use a calculator: For complex calculations, a calculator can help reduce the risk of errors.
- Check your answer: If possible, use a different method or approach to verify your answer.
Alternative Methods for Verification
While the substitution method is straightforward, there are alternative ways to verify if a point is a solution to a system of equations. One such method is graphing.
Graphing the Equations
Each equation in the system represents a line on the coordinate plane. If we graph both lines, the point of intersection (if it exists) represents the solution to the system. To verify if (3, 6) is a solution, we can graph the lines and see if the point lies on both lines.
The first equation, y = (4/3)x + 2, is a linear equation with a slope of 4/3 and a y-intercept of 2. The second equation, y = 3x + 3, is a linear equation with a slope of 3 and a y-intercept of 3.
If we plot these lines on a graph, we can visually inspect whether the point (3, 6) lies on both lines. If the point lies on both lines, it is a solution to the system. If it lies on only one line or neither line, it is not a solution.
Graphing can be a useful way to visualize the solutions to systems of equations, but it may not always provide an exact answer, especially if the point of intersection has non-integer coordinates. In such cases, the substitution method is more reliable for verification.
Conclusion
In conclusion, to determine if (3, 6) is a solution to the system of equations:
y = (4/3)x + 2
y = 3x + 3
We substituted x = 3 and y = 6 into both equations. The point (3, 6) satisfied the first equation but not the second. Therefore, (3, 6) is not a solution to the system of equations.
Understanding how to verify solutions to systems of equations is a crucial skill in algebra and has applications in various fields. By following the step-by-step verification process and being mindful of common mistakes, you can accurately determine whether a given point is a solution. For further learning on systems of equations and related topics, consider exploring resources like Khan Academy's Systems of Equations section.
