How do you solve the following linear system: #-9 y + 2x = 2 , 5x - y = 3#?

Step 1) Solve the second equation for #y#:

#5x - y = 3#

#-color(red)(3) + 5x - y + color(blue)(y) = -color(red)(3) + 3 + color(blue)(y)#

#-3 + 5x - 0 = 0 + color(blue)(y)#

#-3 + 5x = y#

#y = -3 + 5x#

Step 2) Substitute #(-3 + 5x)# for #y# in the first equation and solve for #x#:

#-9y + 2x = 2# becomes:

#-9(-3 + 5x) + 2x = 2#

#(-9 xx -3) + (-9 xx 5x) + 2x = 2#

#27 - 45x + 2x = 2#

#27 + (-45 + 2)x = 2#

#27 - 43x = 2#

#-color(red)(27) + 27 - 43x = -color(red)(27) + 2#

#0 - 43x = -25#

#-43x = -25#

#(-43x)/color(red)(-43) = (-25)/color(red)(-43)#

#(color(red)(cancel(color(black)(-43)))x)/cancel(color(red)(-43)) = 25/43#

#x = 25/43#

Step 3) Substitute #25/43# for #x# in the solution to the second equation at the end of Step 1 and calculate #y#:

#y = -3 + 5x# becomes:

#y = -3 + (5 xx 25/43)#

#y = -3 + 125/43#

#y = (-3 xx 43/43) + 125/43#

#y = -129/43 + 125/43#

#y = -4/43#

The solution is: #x = 25/43# and #y = -4/43# or #(25/43, -4/43)#

From the second eqn., #5x-3=y.#

Subst.ing this value of #y# in the first eqn., we get,

#-9(5x-3)+2x=2.#

#:. -45x+27+2x=2.#

#:. -43x=2-27=-25.#

#:. x=25/43.#

#:. y=5x-3=5(25/43)-3=125/43-3=(125-129)/43.#

#;. y=-4/43.#

Hence, the Soln. #x=25/43, y=-4/43.#