color(brown)("You have 2 unknowns and 2 equations so it is solvable.")You have 2 unknowns and 2 equations so it is solvable.
color(brown)("Two ways of solving:")Two ways of solving:
color(blue)("Method 1")Method 1
In detail: every step shown!!
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Consider -5x-2y=3 −5x−2y=3 making yy the dependant variable
color(brown)("Showing the method for changing the variable for "5x-2y=3" only.")Showing the method for changing the variable for 5x−2y=3 only.
Add color(green)(2y)2y to both sides giving:
(-5x-2y) color(green)(+2y) = (3)color(green)(+2y)(−5x−2y)+2y=(3)+2y
-5x = 3 +2y−5x=3+2y
Subtract color(green)(3)3 from both sides giving:
(-5x) color(green)(-3) =(2y+3)color(green)(-3)(−5x)−3=(2y+3)−3
-5x-3=2y−5x−3=2y
Divide both sides by color(green)(2)2
Note that divide by 2 is the same as multiply by 1/212
color(green)(1/2) (-5x-3) =color(green)(1/2)(2y)12(−5x−3)=12(2y)
-5/2x -3/2 = y−52x−32=y
Write as color(blue)(y=-5/2x-3/2 ....................................(1))
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color(brown)("Changing the variable for "2x-y =-6)
color(blue)(y=2x+6........................................................(2))
Subtract color(green)((1)) from (2) giving:
y - color(green)(y)=2x -color(green)((-5/2x)) +6 -color(green)((-3/2))
0=9/2 x + 15/2
9/2x = -15/2
Multiply throughout by 2 giving:
9x=15
color(blue)(x=15/9................................................(3))
Next step I will leave for you to do!
Substitute (3) into either of (1) or (2) to find the value of y
color(blue)("Method 2")
-5/2x-3/2 =y =2x+6
-5/2x-3/2 =2x+6
Now solve for x and substitute in (1) or (2) to find y
color(red)("Both methods are fast once you are")
color(red)("able to "underline("change the variable by sight")
