Simultaneous Equations

Simultaneous Equations: A Step-by-Step Guide

Simultaneous equations are a set of two or more equations that involve the same unknown variables. Solving them involves finding values for these variables that satisfy all equations simultaneously. This tutorial will guide you through the process using various methods.

1. Substitution Method

Concept:

• Solve one equation for one variable in terms of the other.
• Substitute this expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute the found value back into either original equation to find the value of the first variable.

Example:

Solve the system:

x + y = 5 
2x - y = 4

1. Solve for 'y' in the first equation: y = 5 - x

2. Substitute this value of 'y' into the second equation: 2x - (5 - x) = 4

3. Simplify and solve for 'x': 2x - 5 + x = 4 3x = 9 x = 3

4. Substitute the value of 'x' back into the first equation (or the expression for 'y'): 3 + y = 5 y = 2

Therefore, the solution is x = 3 and y = 2.

2. Elimination Method

Concept:

• Multiply one or both equations by suitable constants to make the coefficients of one variable identical or opposites.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting equation for the remaining variable.
• Substitute the found value back into either original equation to find the value of the eliminated variable.

Example:

Solve the system:

3x + 2y = 11
2x - 3y = -4

1. Multiply the first equation by 3 and the second equation by 2 to make the 'y' coefficients opposites: 9x + 6y = 33 4x - 6y = -8

2. Add the two equations together to eliminate 'y': 13x = 25 x = 25/13

3. Substitute the value of 'x' into either original equation to find 'y': 3(25/13) + 2y = 11 75/13 + 2y = 11 2y = 88/13 y = 44/13

Therefore, the solution is x = 25/13 and y = 44/13.

3. Graphical Method

Concept:

• Graph both equations on the same coordinate plane.
• The point of intersection of the two lines represents the solution to the system of equations.

Example:

Solve the system:

y = 2x + 1
y = -x + 4

1. Graph both equations: [Insert a graph with the two lines intersecting at the point (1, 3)]

2. The point of intersection is (1, 3).

Therefore, the solution is x = 1 and y = 3.

4. Using Matrices

Concept:

• Write the system of equations in matrix form (Ax = b).
• Use matrix operations to solve for the unknown vector x.

Example:

Solve the system:

2x + y = 5
x - 3y = -1

1. Write the system in matrix form: [2 1] [x] = [5] [1 -3] [y] = [-1]

2. Solve for [x] using matrix operations (e.g., inverse matrix or Gaussian elimination).

This method is more advanced and requires knowledge of matrix algebra.

Summary

Simultaneous equations can be solved using various methods. The best approach depends on the specific system and your comfort level with different techniques. Practice and understanding the underlying concepts will make solving these problems more straightforward.