SOLVING SYSTEMS OF LINEAR EQUATIONSAND INEQUALITIES

A SYSTEM OF EQUATIONS is a setof equations with the same variables.

EXAMPLE2x + 3y = 6 x – 2y = 7

What this chapter is about is findingthe solution(s) for these systems.

In the first section (7.1), we will lookat solving systems by graphing.

POSSIBILITIES

LINES INTERSECT AT ONE POINT

ONE SOLUTION – where the lines cross(written as an ordered pair – see blue dot)

THE SYSTEM IS CONSISTENT & INDEPENDENT

LINES ARE PARALLEL

NO SOLUTION(The lines do not cross)

THE SYSTEM IS INCONSISTENT.

LINES COINCIDE (same line)

INFINITE SOLUTIONS(infinite number of points in common)

THE SYSTEM IS CONSISTENT & DEPENDENT.

DEFINITIONS

CONSISTENTA SYSTEM OF EQUATIONS THAT HAS AT LEASTONE ORDERED PAIR THAT SATISFIES BOTH EQUATIONS.

INCONSISTENTA SYSTEM OF EQUATIONS WITH NO ORDERED PAIRTHAT SATISFIES BOTH EQUATIONS

INDEPENDENT

DEPENDENT

A SYSTEM OF EQUATIONS WITH EXACTLY ONE SOLUTION.

A SYSTEM OF EQUATIONS THAT HAS AN INFINITENUMBER OF SOLUTIONS.

RECAP

IF LINES CROSS:

There is one solution.

The system is consistent and independent.

IF LINES ARE PARALLEL:

There Is no solution.

The system is inconsistent.

IF LINES COINCIDE (same line):

There are an infinite number of solutions.

The system is consistent and dependent.

Slopes are different.

Slopes are the same, y-intercepts are different.

Slopes are the same, y-intercepts are same.

PRACTICESOLVE BY GRAPHING

1. y = x + 3 y = -x -1

2. 2x + 3y = 6 2x + 3y = 2

3. 2x - y = 4 y = 2x - 4

PROBLEM #1

SOLUTION: (-2, 1)

1. y = x + 3 y = -x -1

PRACTICESOLVE BY GRAPHING

1. y = x + 3 y = -x -1

2. 2x + 3y = 6 2x + 3y = 2

3. 2x - y = 4 y = 2x - 4

PROBLEM #2

NO SOLUTION

PRACTICESOLVE BY GRAPHING

1. y = x + 3 y = -x -1

2. 2x + 3y = 6 2x + 3y = 2

3. 2x - y = 4 y = 2x - 4

PROBLEM #3

INFINITE SOLUTIONS

YOU CAN FIND THE EXACT SOLUTION OF A SYSTEM BY USING SUBSTITUTION.

SOLVE: y = 2x 2x + 5y = 12

STEP 1: Solve one equation for a variable.In this case, the 1st equation is already solved for y.

STEP 2: Substitute into 2nd equation and solve. 2x + 5y = 12

2x + 5(2x) = 122x + 10x = 12

12x = 12

So, x = 1

YOU CAN FIND THE EXACT SOLUTION OF A SYSTEM BY USING SUBSTITUTION.

SOLVE: y = 2x 2x + 5y = 12

STEP 3: Substitute value in one of the equations.y = 2x

(1, 2)

y = 2(1)

y = 2