Systems of Equations

I. Systems of Linear Equations Four Methods: 1. Elimination by Substitution 2. Elimination by Addition 3. Matrix Method 4. Cramer’s Rule 5. Geometric Method for a 2 by 2 system

Example (1)

Geometric method Graph the lines represented by the equations The solution is the intersection of these lines

Example (2)

Geometric method Graph the lines represented by the equations These lines are parallel and do not intersect, and so no solution for the given system exists.

Example (3)

Geometric method Graph the lines represented by the two equations (they are equivalent equations) representing the same lines

3 by 3 Linear system See the following examples: 1. Example (5) Page Example (6) Page Example (7) Page 146

Cramer’s Rule

Determinants

Two by Two Determinants

Systems of Linear Equations

Two Equations in Two Unknowns

Example

The case when Δ 0 = 0 The left side of the first equation is a k multiple of the left side of the second one, for some real number k The right side of the first equation is a k multiple of the right side of the second one → There are finitely many solutions for the system The right side of the first equation is not a k multiple of the right side of the second one. → There is no solution for the system

Three by Three Determinants

A System of Three Linear Equations in Three Unknowns

Example

Nonlinear System Example (1)

Geometric method Graph the line and the quadratic function represented by the two equations; y = - x 2 + 2x + 7 and y = 3x + 1 The points of their intersection are the solutions of the system

Intersection of Graphs

Example (1)

Why we get two answers, when actually the graphs intersect at only one point?

Answer: Because, when we squared √( x+2), we introduced the other function whose square is also equal x+2 Which function is this? At which point does it intersect the line y = x+2

It is the function y = - √( x+2) It intersects the line y = x+2 at the point whose x coordinate is 7. What is the y-coordinate of this point?