The #"Linear Combination"# method of solving pairs of equations involves adding or subtracting the equations to eliminate one of the variables.
#color(white)(n)##x- y=10#
# 5x+2y=12#
#color(white)(mmmmmmm)##"————————"#
Solve for #x#
1) Multiply all the terms in the first equation by #2# to give both #y# terms the same coefficients
#color(white)(.)##2x -2y=20#
2) Add the second equation to the doubled one in order to make the #2y# terms go to #0# and drop out
#color(white)(.n)##2x-2y=20#
#+ 5x+2y=12#
#"————————"#
#color(white)(.n)##7x # #color(white)(.n...)# #= 32#
3) Divide both sides by #7# to isolate #x#
#x = (32)/(7)# #larr# answer for #x#
#color(white)(mmmmmmm)##"————————"#
Solve for #y#
1) Sub in to one of the original equations the value of #x# and solve for #y#
#color(white)(.)##x - y = 10#
#(32)/(7) - y = 10#
2) Clear the denominator by multiplying all the terms on both sides by #7# and letting the denominator cancel
#32 - 7y = 70#
3) Subtract #32# from both sides to isolate the #-7y# term
#-7y = 38#
4) Divide both sides by #-7# to isolate #y#
#y = - (38)/(7)# #larr# answer for #y#
#color(white)(mmmmmmm)##"————————"#
Answer
#x = (32)/(7)#
#y = - (38)/(7)#
#color(white)(mmmmmmm)##"————————"#
Check
Sub in the values to see if the equation is still true.
# 5x+2y=12#
#((5)/(1)xx(32)/(7))# #+ ((2)/(1)xx (-38)/(7))# should equal #12#
#(160)/(7) - (76)/(7)# should equal #12#
#(84)/(7)# should equal #12#
#12# does equal #12#
#Check!#
