Page 2: Advanced Applications and Practice Problems

The second page expands on the concept with more complex practice problems for solving systems of equations by substitution, including special cases and various equation formats. It demonstrates how the substitution method can handle different types of systems, from standard linear equations to more complex variations.

Highlight: Special cases are explored, including parallel lines (no solution) and coincident lines (infinite solutions).

Example: The system {y=x+6, y=-2x-3} is solved to find the intersection point (-3,3), demonstrating how substitution can efficiently find points of intersection.

Vocabulary: Parallel lines are indicated by the symbol Ø (no solution), while coincident lines have infinitely many solutions.

Definition: A solution to a system of equations is a point (x,y) that satisfies all equations in the system simultaneously.

Page 1: Solving Systems by Substitution

This page introduces the fundamental steps of solving systems of equations through substitution, providing detailed examples of the process. The method is systematically broken down into three main steps, followed by multiple practice problems demonstrating various applications.

Definition: The substitution method is a technique for solving systems of equations by expressing one variable in terms of another and substituting this expression into the second equation.

Highlight: The three essential steps for solving systems by substitution are:

1. Solve one equation for x or y
2. Substitute this expression into the other equation
3. Substitute the solution back to find the other variable

Example: For the system {4x-1=2x-5, y=4x-1}, solving yields the solution point (-2,-9) through systematic substitution.

Vocabulary: A system of equations consists of two or more equations that must be solved simultaneously to find values that satisfy all equations.