Question

The difference of two numbers, a and b, is 21. The difference of 5 times a and two times b is 18. What are the values of a and b?

Answer 1

\[a-b=21\]
\[5a-2b=18\]
we can use substitution
\[a-b=21\]
\[a=21+b\]
Substitute in the second equation
\[5(21+b)-2b=18\]
\[105+5b-2b=18\]
\[3b=-87\]
\[b=-29\]
Go back to first equation
\[a-b=21\]
\[a--29=21\]
\[a+29=21\]
\[a=-8\]

Answer 2

given a - b = 21
given 5 a - 2 b = 18
solve simultaneously by substitution or by elimination
to use substitution:
from the first equation let a = 21 + b
then 5(21 + b ) - 2 b = 18
105 + 5 b - 2 b = 18
b = -87/3 = -29
a = 21- 29 = -8

Answer 3

to use elimination
multiply the first equation by -2 add the second
-2 a + 2 b = -42
5 a -2 a + 2 b - 2 b = -42 + 18
3 a = -24
a = -8
substitute for a in original equation to get b