Solving Systems of Equations Algebraically

Solve the following system using elimination: 2x - 2y = 10 and 3x + 2y = 20

Solve the following system using elimination: 4x - y = 5 and x + 5y = -4

Solve the following system using elimination: 3x + 5y = 1 and 4x + 2y = -8

Solve the following system using substitution: x - 4y = 1 and 3x - y = 14

Solve the following system using substitution: 2x - 4y = 6 and 3x + 9y = 9

Solve the following system: y = x² + 2x - 3 and -3x + y = -1

Solve the following system: x² + y² = 17 and y = x - 3

Solving Systems of Equations

Systems of equations can be solved using elimination or substitution methods.

Elimination Method

The elimination method involves manipulating equations to eliminate one variable, making it easier to solve for the other.
Example: For the system
- 2x - 2y = 10
- 3x + 2y = 20
- The solution is (6, 1).
Another system using elimination:
- 4x - y = 5
- x + 5y = -4
- The solution is (1, -1).
A third system also solved by elimination:
- 3x + 5y = 1
- 4x + 2y = -8
- The solution is (-3, 2).

Substitution Method

The substitution method requires isolating one variable in terms of the other and substituting it into the second equation.
Example system:
- x - 4y = 1
- 3x - y = 14
- The solution is (5, 1).
Another system solved by substitution:
- 2x - 4y = 6
- 3x + 9y = 9
- The solution is (3, 0).

Mixed Methods

Systems can also combine quadratic and linear equations.
Example:
- y = x² + 2x - 3
- -3x + y = -1
- Solutions are (2, 5) and (-1, 4).
Another unique system with a circle and line:
- x² + y² = 17
- y = x - 3
- Solutions are (4, 1) and (-1, -4).

Summary

Understanding both elimination and substitution methods is essential for solving systems of equations efficiently.
Solutions can vary widely, including multiple points for non-linear systems.

Test your knowledge on solving systems of equations using elimination and substitution methods. This quiz covers various examples and techniques to help reinforce your understanding of algebraic concepts. Challenge yourself with problems that require critical thinking and problem-solving skills.