# Intermediate Algebra

Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
10x + 7y = -120.5
-20x – 14y = 241

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1. There are three possible cases when we have a system of two equations in two unknowns.
1. There is a unique solution.
2. There is an infinite number of solutions.
3. There is no solution.

Case 1:
Solution can be found by elimination or substitution.
Example:
x+2y=3 ...(1)
x-y =0 ...(2)
Solve by elimination:
(1)-(2)
3y = 3
y=1
Substitute y=1 in (2) to get x=1.

Case 2.
When one equation is a linear combination of the other, we essentially have one equation for two unknowns. There is an infinite set of solutions.
Example:
x+2y = 3
3x+6y = 9

case 3.
When the left-hand-side of one equation is the linear combination of the other, but the right hand side of the equations are not the same, then the equations are not consistent and there is no solution.
Example:
2x+4y = 6
x+2y = 0

See if you can identify to which case the given system belongs.

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