Substitution Method Examples

This page demonstrates the substitution method for solving systems of equations with step-by-step solutions.

The first example solves the system: 3x + 2y = 27 2x - y = 0

The solution process involves:

1. Isolating y in the second equation
2. Substituting the expression for y into the first equation
3. Solving for x
4. Back-substituting to find y
5. Verifying the solution

Example: The solution (x, y) = (1, 2) is obtained through careful substitution and algebraic manipulation.

The second example tackles: 3x + 8y = 7 4x + (-2) = 0

This problem showcases a slightly different approach, highlighting the flexibility of the substitution method.

Highlight: The substitution method is particularly useful when one variable can be easily isolated in one of the equations.

Elimination Method and Special Cases

This page covers the elimination method for solving systems of equations and explores special cases.

The elimination method is demonstrated with the system: x + 4y = 2 -x + y = 8

Key steps include:

1. Aligning equations to eliminate one variable
2. Adding or subtracting equations to cancel out a variable
3. Solving for the remaining variable
4. Back-substituting to find the other variable

Example: The solution (-6, 2) is obtained by eliminating x and then solving for y.

The page also introduces a special case where the system has infinitely many solutions: 4x + 2y = 12 8x + 4y = 24

Highlight: When equations in a system are multiples of each other, they represent the same line, resulting in infinitely many solutions.

Definition: Infinitely many solutions occur when the equations in a system are equivalent, representing the same line graphically.

This concept is crucial for understanding the nature of linear systems and their graphical representations.