Question

How do you solve the following linear system: # - 12x + y = 15 , 36x + 22y = 3 #?

Answer 1

See a solution process below:

Explanation:

Step 1) Solve both the equations for `36x`

• Equation 1:

`-12x + y = 15`

`color(red)(-3)(-12x + y) = color(red)(-3) xx 15`

`(color(red)(-3) xx -12x) + (color(red)(-3) xx y) = -45`

`36x + (-3y) = -45`

`36x - 3y = -45`

`36x - 3y + color(red)(3y) = -45 + color(red)(3y)`

`36x - 0 = -45 + 3y`

`36x = -45 + 3y`

• Equation 2:

`36x + 22y = 3`

`36x + 22y - color(red)(22y) = 3 - color(red)(22y)`

`36x + 0 = 3 - 22y`

`36x = 3 - 22y`

Step 2) Because the left side of both equations are the same we can equate the right side of each equation and solve for `y`:

`-45 + 3y = 3 - 22y`

`-45 + color(red)(45) + 3y + color(blue)(22y) = 3 + color(red)(45) - 22y + color(blue)(22y)`

`0 + (3 + color(blue)(22))y = 48 - 0`

`25y = 48`

`(25y)/color(red)(25) = 48/color(red)(25)`

`(color(red)(cancel(color(black)(25)))y)/cancel(color(red)(25)) = 48/25`

`y = 48/25`

Step 3) Substitute `48/25` for `y` in the solution to either equation in Step 1 and solve for `x`:

`36x = 3 - 22y` becomes:

`36x = 3 - (22 xx 48/25)`

`36x = (25/25 xx 3) - (22 xx 48/25)`

`36x = 75/25 - 1056/25)`

`36x = -981/25`

`1/36 xx 36x = 1/36 xx -981/25`

`36/36x = -981/900`

`x = -981/900`

The Solution Is:

`x = -981/900` and `y = 48/25`

Or

`(-981/900, 48/25)`

Answer 2

`x=-1.09, y=1.92`

Explanation:

`-12x+y=15`-----(1)
`36x+22y=3`-----(2)
`-36x+3y=45`----(3)=3(1)
`25y=48`-------(4)=(2)+(3)
`y=48/25=1.92`-------(5)=(4)/25
`-12x+y=15`-----(1)
`-12x+1.92=15" "`Substitute for y
`-12x=15-1.92" "`Transpose 1.92
`-12x=13.08`
`x=13.08/-12=-1.09`

Thus,
`x=-1.09, y=1.92`
`-12xx(-1.09)+1.92=15`Checked
`36xx(-1.09)+22xx1.92=3`Checked