Y 6x 11 2x 3y 7

Decoding the Mathematical Puzzle: y = 6x + 11, 2x + 3y = 7

This article delves into the intricacies of solving the simultaneous equations y = 6x + 11 and 2x + 3y = 7. We'll explore various methods, understand the underlying concepts, and demonstrate how to approach similar problems. This comprehensive guide is designed for students, educators, and anyone interested in strengthening their mathematical problem-solving skills.

Understanding Simultaneous Equations

Simultaneous equations, also known as systems of equations, involve two or more equations with two or more variables. The goal is to find values for the variables that satisfy all equations simultaneously. These equations often represent relationships between different quantities, and finding the solution reveals the point where these relationships intersect. In our case, we have a system of two linear equations:

• Equation 1: y = 6x + 11
• Equation 2: 2x + 3y = 7

These equations represent two straight lines on a graph. The solution to the system is the coordinates (x, y) of the point where these two lines intersect.

Method 1: Substitution Method

The substitution method is a straightforward approach for solving simultaneous equations. It involves solving one equation for one variable and substituting the resulting expression into the other equation.

1. Solve for one variable: Equation 1 is already solved for 'y': y = 6x + 11

2. Substitute: Substitute the expression for 'y' (6x + 11) into Equation 2:

   2x + 3(6x + 11) = 7

3. Simplify and solve for x:

   2x + 18x + 33 = 7 20x = 7 - 33 20x = -26 x = -26/20 = -13/10

4. Substitute back to find y: Now, substitute the value of x (-13/10) back into Equation 1 to find y:

   y = 6(-13/10) + 11 y = -78/10 + 110/10 y = 32/10 = 16/5

Therefore, the solution using the substitution method is x = -13/10 and y = 16/5.

Method 2: Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations to eliminate one variable by adding or subtracting them.

1. Prepare the equations: We need to make the coefficients of either x or y opposites. Let's multiply Equation 1 by -3 to eliminate y:

   -3(y) = -3(6x + 11) => -3y = -18x - 33

2. Add the equations: Add the modified Equation 1 to Equation 2:

   2x + 3y = 7 -3y = -18x - 33

   2x - 3y + 3y = 7 - 18x - 33 2x = -18x - 26

3. Solve for x:

   20x = -26 x = -13/10

4. Substitute to find y: Substitute the value of x (-13/10) back into either Equation 1 or 2 to find y. Using Equation 1:

   y = 6(-13/10) + 11 y = 16/5

Again, the solution is x = -13/10 and y = 16/5.

Graphical Representation

Plotting these equations on a graph provides a visual representation of the solution. The point of intersection of the two lines represents the solution (x, y). While we won't create a graph here, you can easily do so using graphing software or by hand. The coordinates of the intersection point should confirm our calculated solution: x = -13/10 and y = 16/5.

Verifying the Solution

To ensure accuracy, it's crucial to verify the solution by substituting the values of x and y into both original equations.

Equation 1: y = 6x + 11

16/5 = 6(-13/10) + 11 16/5 = -78/10 + 110/10 16/5 = 32/10 16/5 = 16/5 (Correct)

Equation 2: 2x + 3y = 7

2(-13/10) + 3(16/5) = 7 -26/10 + 96/10 = 7 70/10 = 7 7 = 7 (Correct)

Both equations are satisfied, confirming the accuracy of our solution.

Applications of Simultaneous Equations

Simultaneous equations have numerous applications in various fields:

• Physics: Solving problems related to motion, forces, and electricity often requires solving simultaneous equations.
• Engineering: Designing structures, analyzing circuits, and modeling systems frequently involve simultaneous equations.
• Economics: Analyzing market equilibrium, supply and demand, and optimizing resource allocation often uses systems of equations.
• Computer Science: Solving algorithms, optimizing network flows, and computer graphics all utilize these mathematical techniques.

Expanding on the Problem: Exploring Variations

The problem y = 6x + 11 and 2x + 3y = 7 is a fundamental example. Let's explore how variations in the equations might affect the solution:

• Parallel Lines: If the equations represented parallel lines, there would be no solution, as the lines never intersect. This occurs when the equations have the same slope but different y-intercepts. For example, y = 6x + 11 and y = 6x + 5.

• Coincident Lines: If the equations represented the same line (coincident lines), there would be infinitely many solutions. This happens when one equation is a multiple of the other. For example, y = 6x + 11 and 2y = 12x + 22.

• Nonlinear Equations: If the equations involved higher-order terms (e.g., x², y²), the solution methods would become more complex, possibly requiring techniques such as substitution, elimination, or graphical methods to find the points of intersection.

Conclusion

Solving simultaneous equations like y = 6x + 11 and 2x + 3y = 7 is a fundamental skill in mathematics with wide-ranging applications. Understanding both the substitution and elimination methods provides flexibility in tackling various types of problems. By carefully following the steps, verifying the solution, and understanding the graphical representation, one can build a solid foundation in solving simultaneous equations and apply this knowledge to complex real-world problems. Remember to always verify your solutions to ensure accuracy and to explore variations of the problem to enhance your understanding of the underlying mathematical principles. This deeper understanding will prove invaluable in more advanced mathematical studies and real-world applications.