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SOLVING SYSTEMS OF
LINEAR EQUATIONS
AND INEQUALITIES
A SYSTEM OF EQUATIONS is a set
of equations with the same variables.
EXAMPLE
2x + 3y = 6
x – 2y = 7
What this chapter is about is finding
the solution(s) for these systems.
In the first section (7.1), we will look
at solving systems by graphing.
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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
CONSISTENT
A SYSTEM OF EQUATIONS THAT HAS AT LEAST
ONE ORDERED PAIR THAT SATISFIES BOTH EQUATIONS.
INCONSISTENT
A SYSTEM OF EQUATIONS WITH NO ORDERED PAIR
THAT SATISFIES BOTH EQUATIONS
DEPENDENT
A SYSTEM OF EQUATIONS THAT HAS AN INFINITE
NUMBER OF SOLUTIONS.
INDEPENDENT
A SYSTEM OF EQUATIONS WITH EXACTLY ONE SOLUTION.
IF LINES CROSS:
There is one solution.
The system is consistent and independent.
Slopes are different.
IF LINES ARE PARALLEL:
There Is no solution.
The system is inconsistent.
Slopes are the same, y-intercepts are different.
IF LINES COINCIDE (same line):
There are an infinite number of solutions.
The system is consistent and dependent.
Slopes are the same, y-intercepts are same.
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PRACTICE
SOLVE 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)
PRACTICE
SOLVE 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
PRACTICE
SOLVE 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) = 12
So, x = 1
2x + 10x = 12
12x = 12
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
y = 2(1)
y=2
(1, 2)