Question

How do you solve the system of equations `2x + y = 6` and `- 2x - y = 6`?

Answer 1

There is no solution to this system of equations

Explanation:

`color(purple)("For there to be a solution the lines have to cross at some point.")`

Given:`" "2x+y=6" "...................Equation(1)`
`" "-2x-y=6" "...................Equation(2)`

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`color(blue)("Consider equation(1)")`

Subtract `color(red)(2x)` from both sides

`color(green)(2x+y=6" "->" "2xcolor(red)(-2x) +y=color(red)(-2x)+6`

`" "0+y=-2x+6`
`" "y=-2x+6..Equation(1_a)`

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`color(blue)("Consider equation(2)")`

Add `y` to both sides`" "->" "-2x=6+y`

Subtract 6 from both sides`->" "-2x-6=y`

Write as: `" "->" "y=-2x-6..Equation(2_a)`

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Compare both `Equation(1_a)" and "Equation(2_a)` to the standard straight line equation form of `y=mx+c`

Note that `m` is the slope (gradient) of the line

This means that both equations have the same slope of -2 (from `-2x`) thus they are parallel.

Now consider the value of `c" in "y=mx+c`
In one equation it is +6 and in the other it is -6.

So they cross the y-axis in different places. Thus the lines never cross each other and as a consequence do not share values. So there is no solution to this system of equations.