If 3a+2b=24 and 4a+5b=53 what is the value of a+b

3a + 2b = 24
4a + 5b = 53

To use the addition method, we can multiply the first equation by 5 and the second equation by -2 to get b-terms with opposite coefficients:

5(3a + 2b = 24)      -->      15a + 10b = 120
-2(4a + 5b = 53)      -->      -8a -10b = -106

Add the equations together and the b-terms cancel out:

  15a + 10b = 120
    -8a - 10b = -106
+ 
               7a = 14
                 a = 2

To find b, substitute back using a:

3(2) + 2b = 24
6 + 2b = 24
2b = 18
b = 9

Finally, we must be sure to check our work by substituting back:

3(2) + 2(9) = 24    -->    6 + 18 = 24    <--True
4(2) + 5(9) = 53    -->    8 + 45 = 53    <--True

a = 2
b = 9

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