3x 2y 2 5x 5y 10

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

This article delves into the fascinating world of simultaneous equations, specifically focusing on the relationship between the two linear equations: 3x + 2y = 2 and 5x + 5y = 10. We'll explore various methods to solve this system, analyze the geometric interpretation, and discuss the broader implications of such problems in mathematics and beyond. Understanding these concepts is crucial for anyone studying algebra, and the principles extend to numerous applications in fields like physics, engineering, and economics.

Understanding Simultaneous Equations

Simultaneous equations, also known as systems of equations, involve two or more equations with the same set of unknowns (variables). The goal is to find values for these unknowns that satisfy all equations simultaneously. In our case, the unknowns are x and y, and we need to find the values of x and y that make both 3x + 2y = 2 and 5x + 5y = 10 true.

Types of Simultaneous Equations

Simultaneous equations can be classified into different types based on their form and the number of solutions:

• Linear Equations: These are equations where the variables are raised to the power of 1. Both equations in our example are linear.
• Non-linear Equations: These involve variables raised to powers other than 1 (e.g., quadratic equations, cubic equations).
• Consistent Systems: These systems have at least one solution. Our system may or may not be consistent, which we will determine.
• Inconsistent Systems: These systems have no solutions. The equations represent parallel lines that never intersect.
• Dependent Systems: These systems have infinitely many solutions. The equations represent the same line.

Solving the Simultaneous Equations: Method 1 - Elimination

The elimination method involves manipulating the equations to eliminate one variable, allowing us to solve for the other. Let's apply this to our system:

1. Simplify the second equation:

The second equation, 5x + 5y = 10, can be simplified by dividing both sides by 5:

x + y = 2

2. Solve for one variable in terms of the other:

Let's solve the simplified equation for x:

x = 2 - y

3. Substitute into the first equation:

Substitute this expression for x into the first equation, 3x + 2y = 2:

3(2 - y) + 2y = 2

4. Solve for y:

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

5. Substitute back to find x:

Now substitute the value of y (4) back into either of the original equations or the simplified equation (x = 2 - y) to find x:

x = 2 - 4 x = -2

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

Solving the Simultaneous Equations: Method 2 - Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. We've essentially done this in the elimination method above, but let's demonstrate a slightly different approach.

1. Solve one equation for one variable:

Let's solve the simplified equation (x + y = 2) for y:

y = 2 - x

2. Substitute into the other equation:

Substitute this expression for y into the first equation (3x + 2y = 2):

3x + 2(2 - x) = 2

3. Solve for x:

3x + 4 - 2x = 2 x = -2

4. Substitute back to find y:

Substitute the value of x (-2) back into the expression for y (y = 2 - x):

y = 2 - (-2) y = 4

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

Solving the Simultaneous Equations: Method 3 - Graphical Method

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

1. Rearrange equations into slope-intercept form:

Rewrite both equations in the form y = mx + c, where m is the slope and c is the y-intercept:

For 3x + 2y = 2: 2y = -3x + 2 y = (-3/2)x + 1

For 5x + 5y = 10: 5y = -5x + 10 y = -x + 2

2. Plot the lines:

Plot both lines on a graph. You'll find they intersect at the point (-2, 4).

3. Identify the intersection point:

The coordinates of the intersection point represent the solution: x = -2 and y = 4. This visually confirms the solution obtained through algebraic methods.

Geometric Interpretation

Geometrically, each linear equation represents a straight line on a Cartesian plane. The solution to the simultaneous equations is the point where these two lines intersect. If the lines are parallel, there is no solution (inconsistent system). If the lines are coincident (the same line), there are infinitely many solutions (dependent system). In our case, the lines intersect at a single point, indicating a unique solution.

Applications of Simultaneous Equations

Simultaneous equations are fundamental to numerous applications across various fields:

• Physics: Solving problems involving forces, motion, and electricity often requires solving simultaneous equations.
• Engineering: Designing structures, analyzing circuits, and optimizing processes frequently involve simultaneous equations.
• Economics: Modeling supply and demand, analyzing market equilibrium, and forecasting economic trends often utilize simultaneous equations.
• Computer Science: Solving systems of equations is essential in computer graphics, machine learning, and optimization algorithms.
• Chemistry: Determining the concentration of solutions or balancing chemical reactions often requires solving simultaneous equations.

Conclusion

Solving the simultaneous equations 3x + 2y = 2 and 5x + 5y = 10 has demonstrated the versatility of different algebraic techniques, including elimination, substitution, and the graphical method. Understanding these methods provides a strong foundation for tackling more complex systems of equations in various contexts. The geometric interpretation offers valuable insights into the nature of solutions and the relationships between the equations. The widespread applications of simultaneous equations highlight their importance in diverse fields, reinforcing their significance in mathematical and scientific problem-solving. The unique solution (x = -2, y = 4) underlines the consistent nature of this particular system of equations. Further exploration of more complex systems, including those involving non-linear equations, would enhance one's understanding of this powerful mathematical tool.