How do you solve the following system: # 3x – 5y = 53 ,5x - 7y = 12 #?

2 Answers
Mar 9, 2017

Answer:

Why on earth do they chose such awful numbers????

#x=-77.75#
#y=-57.25#

Explanation:

In solving equations for variable values the method is to manipulate so that you have a load of values and just one variable. It is then solvable. Variables in this case being #x and y#.

#3x-5y=53" "....................Equation(1)#
#5x-7y=12" "....................Equation(2)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving for "x)#

Consider equation(1)

To make the -5y positive multiply everything by -1

#-3x+5y=-53#

Add #3x# to both sides

#color(green)(-3xcolor(red)(+3x)+5y=-53color(red)(+3x))#

But #-3x+3x=0#

#0+5y=3x-53#

Divide both sides by 5 ( this is the same as #color(red)(xx1/5)" "#)

#color(green)(5/(color(red)(5))y=3/(color(red)(5))x-53/(color(red)(5))#

But #5/5=1#

#y=3/5x-53/5" ".....................Equation(1_a)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Using #Equation(1_a)# substitute for y in #Equation(2)#

#5x-7y=12" "->" "5x-7(3/5x-53/5)=12#

#" "25/5x" "-21/5x+ 371/5=12#

#" "4/5x=12-371/5#

#" "4x=60-371#

#" "color(blue)(x= -311/4)#

#color(blue)("This is the same as "x=-77.75)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Solving for "y)#

Substitute for #x# in #Equation(2)#
It does not matter which of the 2 equations you use.

#5x-7y=12" "->" "5(-311/4)-7y=12#

#" "-1555/4-7y=12#

#" "-1555/4-28/4y=48/4#

#" "28y=-1555-48#

#" "y= -229/4#

#color(blue)("This is the same as "y=-57.25)#

Mar 9, 2017

Answer:

#x = -77.75 and y = -57.25#

Explanation:

There are several methods for solving a system of equations, also known as simultaneous equations. The method you choose will depend on the format of the equations you are given.

  • Elimination - [make additive inverses and add the equations]
  • Substitution - [write one variable in terms of the other]
  • Equating - [one variable is expressed in two different ways]
  • Matrices - [ multiply a matrix by its inverse to get the unit matrix]
  • Graphically - [ graph each equation and find the intersection]

In this example I will use the elimination method.

#color(white)(.....................)3x -5y = 53color(white)(.............)A#
#color(white)(.....................)5x -7y = 12color(white)(.............)B#

#A xx 5:" "color(red)(15x)-25y = 265color(white)(..........)C#
#B xx"-3: "color(red)(-15x)+21y = -36color(white)(.......)D#

#C+D:color(white)(.............)-4y = 229" "# there is no x-term!
#color(white)(...............................)color(blue)(y = -57.25)#

Now that you have value for #color(blue)(y)#, substitute it into any of the equations above to find #x#

#color(white)(.....................)5x -7color(blue)(y) = 12color(white)(.............)B#

#color(white)(.............)5x -7color(blue)(("-57.25")) = 12#
#color(white)(...................)5x color(blue)("+400.75") = 12#
#color(white)(....................)5x " "= -388.75#
#color(white)(......................)x " "= -77.75#
.

[Note that: #color(red)(15x) and color(red)(-15x)# are additive inverses. They therefore ADD to give 0.]

This was done by multiplying B by #-3# to change the sign.