3x 2y 2 5x 5y 10

Decoding the Mystery: Exploring the Mathematical Relationship Between 3x + 2y = 2 and 5x + 5y = 10

This article delves into the fascinating world of simultaneous linear equations, specifically focusing on the system presented by 3x + 2y = 2 and 5x + 5y = 10. We'll explore various methods to solve this system, analyze its geometric interpretation, and discuss the implications of different solution approaches. Understanding this seemingly simple system reveals fundamental concepts crucial for higher-level mathematics and applications in various fields.

Understanding Simultaneous Linear Equations

Before we tackle the specific equations, let's establish a solid understanding of simultaneous linear equations. These equations involve two or more linear equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. In our case, we have two equations with two variables, x and y.

• Linear Equation: A linear equation is an equation that represents a straight line when graphed. It has the general form Ax + By = C, where A, B, and C are constants, and x and y are variables.

• Simultaneous Solution: The simultaneous solution of a system of linear equations is the set of values for the variables that make all equations true. This system can have one solution, infinitely many solutions, or no solution at all.

Method 1: Solving by Elimination

The elimination method involves manipulating the equations to eliminate one variable, leaving a single equation with one variable that can be easily solved. Let's apply this method to our system:

3x + 2y = 2 (Equation 1) 5x + 5y = 10 (Equation 2)

First, we simplify Equation 2 by dividing both sides by 5:

x + y = 2 (Equation 2 simplified)

Now, we can use this simplified equation to eliminate one variable. Let's eliminate 'x'. Multiply Equation 2 simplified by -3:

-3x - 3y = -6 (Equation 2 multiplied by -3)

Add this modified Equation 2 to Equation 1:

(3x + 2y) + (-3x - 3y) = 2 + (-6) -y = -4 y = 4

Now that we have the value of y, we can substitute it back into either Equation 1 or Equation 2 simplified to solve for x. Let's use Equation 2 simplified:

x + 4 = 2 x = -2

Therefore, the solution to the system of equations is x = -2 and y = 4.

Method 2: Solving by Substitution

The substitution method involves solving one equation for one variable in terms of the other, and then substituting this expression into the other equation. Let's apply this method to our system:

3x + 2y = 2 (Equation 1) 5x + 5y = 10 (Equation 2)

Again, let's simplify Equation 2:

x + y = 2 (Equation 2 simplified)

Now, let's solve Equation 2 simplified for x:

x = 2 - y

Substitute this expression for x into Equation 1:

3(2 - y) + 2y = 2 6 - 3y + 2y = 2 6 - y = 2 y = 4

Substitute the value of y back into x = 2 - y:

x = 2 - 4 x = -2

Once again, we arrive at the solution x = -2 and y = 4.

Method 3: Graphical Solution

A graphical solution involves plotting both equations on a Cartesian coordinate system. The point where the two lines intersect represents the solution to the system.

To graph each equation, we can find the x and y-intercepts.

For 3x + 2y = 2:

• x-intercept (set y = 0): 3x = 2 => x = 2/3
• y-intercept (set x = 0): 2y = 2 => y = 1

For x + y = 2:

• x-intercept (set y = 0): x = 2
• y-intercept (set x = 0): y = 2

Plotting these points and drawing the lines, we observe that they intersect at the point (-2, 4), confirming our previous solutions.

Analyzing the Solution and its Implications

The solution x = -2 and y = 4 is unique to this system. This means there is only one set of values for x and y that satisfies both equations simultaneously. This is because the two lines representing the equations intersect at a single point.

If the lines were parallel (having the same slope but different y-intercepts), the system would have no solution. If the lines were coincident (representing the same line), the system would have infinitely many solutions.

Applications of Simultaneous Equations

Simultaneous equations are used extensively in various fields:

• Engineering: Solving for forces in structures, analyzing circuits, and modeling dynamic systems.
• Physics: Determining projectile motion, analyzing fluid dynamics, and modeling oscillatory systems.
• Economics: Analyzing supply and demand, optimizing resource allocation, and forecasting economic trends.
• Computer Science: Solving systems of linear equations is crucial in computer graphics, machine learning, and optimization algorithms.

Expanding the Understanding: Matrix Representation and Beyond

The system can also be represented using matrices:

[ 3  2 ] [ x ] = [ 2 ]
[ 5  5 ] [ y ] = [ 10]

This matrix representation allows for the application of more advanced techniques like Gaussian elimination or Cramer's rule for solving larger systems of equations. These methods are particularly useful when dealing with a high number of variables and equations.

Conclusion

Solving the system of equations 3x + 2y = 2 and 5x + 5y = 10 demonstrates fundamental concepts in algebra and linear systems. We explored three different methods – elimination, substitution, and graphical representation – each providing a unique perspective on the solution. The unique solution (x = -2, y = 4) highlights the importance of understanding the relationship between the equations and their geometric interpretation. The wide range of applications underscores the significance of mastering these techniques for various fields of study and professional endeavors. The ability to solve simultaneous equations is a cornerstone of mathematical literacy and problem-solving skills that extend far beyond the classroom. By exploring these concepts thoroughly, you're building a strong foundation for more complex mathematical challenges ahead.