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Question 1014327: For what values of the coefficients a and b is the ordered pair (2,1) the solution to the linear system ax+by=-7 and 2ax-3by=1?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your solution is (2,1).
this means x = 2 and y = 1
your equations are:
ax + by = -7
2ax - 3by = 1
since you know the value of x and y, you can replace x and y with their values in those equations.
when x = 2 and y = 1, the equations of:
ax + by = -7
2ax - 3by = 1
become:
2a + 1b = -7
4a - 3b = 1
the two equations you have to work with are now:
2a + 1b = -7
4a - 3b = 1
multiply both sides of the first equation by 2 and leave the second equation as is to get:
4a + 2b = -14
4a - 3b = 1
subtract the second equation from the first equation to get:
5b = -15
solve for b to get b = -3
in the first original equation, solve for a to get:
first original equation is 2a + 1b = -7
when b = -3, this equation becomes 2a - 3 = -7
add 3 to both sides of the equation to get 2a = -4
solve for a to get a = -2
you have a = -2 and b = -3
go back to your original equations and replace a with -2 and b with -3.
your original equations are:
ax + by = -7
2ax - 3by = 1
when a = -2 and b = -3, these equations become:
-2x -3y = -7
-4x + 9y = 1
to confirm that your values of a and b are correct, solve these two equations for x and y.
you should get x = 2 and y = 1
start with:
-2x - 3y = -7
-4x + 9y = 1
multiply both sides of the first equation by 2 and leave the second equation as is to get:
-4x - 6y = -14
-4x + 9y = 1
subtract the second equation from the first to get:
-15y = -15
solve for y to get y = 1.
go back to your original first equation and solve for x.
your first original equation in this series is -2x - 3y = -7
when y = 1, this equation becomes -2x - 3 = -7
add 3 to both sides of this equation to get -2x = -4
divide both sides of this equation by -2 to get x = 2
you get x = 2 and y = 1 as you should, since that's the value of x and y that you started with when solving for a and b.
your solution is:
a = -2
b = -3
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