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what is the answer to a+7b=13 and a+4b=4

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2 Answers

The elimination method requires the coefficient (number multiplied by the variable) to be the same for one variable in both equations.  Here, the invisible 1 in front of the a allows us to use elimination.  When the signs are the same, we subtract; when different, we add.  The 1a is positive in both equations, so we can subtract one entire equation from another to ELIMINATE the variable a.  

     a + 7b = 13     
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     a + 4b = 4     <-- Make sure you subtract each of the terms in the second equation from the first.
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           3b = 9   <-- the a's canceled, 7b - 4b = 3b   and    13 - 4 = 9

Now, we solve by dividing both sides of the equation by 3.

     3b = 9
     3      3

       b = 3

 

So, we know b = 3 and to solve for a, we substitute b back into EITHER equation to solve for a.

 

a + 7b = 13

a + 7(3) = 13

a + 21 = 13
   -  21    -21
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a = -8

 

So our solution, (a,b) can be written as (-8, 3).

    a + 7b = 13
 – 
    a + 4b =  4    
    0 + 3b =  9

    3b = 9 ---> b = 3
In second equation, let's substitute "b" by its value
    a + 4 • 3 = 4
    a = 4 - 12 ---> a = - 8
The solution (a, b) of a system of two linear equations is (- 8, 3)
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Check the answer
1. - 8 + 7 · 3 = 13
        13 = 13
2. - 8 + 4 · 3 = 4
          4 = 4