Answer:
Look at the variables and numbers in the equation, and what it equals
Step-by-step explanation:
Expressions that are the exact same have infinite solutions
[tex]5x+3=5x+3[/tex]
Expressions that are different have 1 solution
[tex]6x+7=5x-2[/tex]
Expressions that are the same with different operands have no solution
[tex]5x-3=5x+3[/tex]
Explanation:
The short of it is that when the equations are in the same form, if one equation is the same as or a multiple of the other, there are infinite solutions. If one equation is the same as or a multiple of the other, except for the constant terms, there are no solutions. Otherwise, there is one solution.
Example:
If one equation is x +2y = 3, then you can consider the other equation:
3x +6y = 9 . . . . 3 times the first equation: infinite solutions
3x +6y = 6 . . . . 3 times the first equation; constants differ: no solutions
3x -6y = 9 . . . . not a multiple of the first equation: one solution
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Here's a more formal method.
Put both equations of the linear system of two equations into general form:
ax +by +c = 0
dx +ey +g = 0
Perform the following arithmetic:
p = ae -db . . . . . this is called the system "determinant"
q = bg -ec
r = cd -ga
If p = q = r = 0, the equations describe the same line, so there are infinite solutions.
If p = 0 and either of q ≠ 0 or r ≠ 0, the equations describe parallel lines, so there are no solutions.
If p ≠ 0, there is one solution: (x, y) = (q/p, r/p).
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Additional comment
This method can always be used to find the solution to a system that is known to have one solution. It often requires fewer arithmetic operations than other solution methods. It is a variation of what is sometimes called the "cross multiplication method."
