Question

How do solve the following linear system?: `-7x+y=-19 , -2x+3y=-1`?

Answer 1

`(x,y)=(56/19,31/19)`

Explanation:

`{(-7x+y=-19),(-2x+3y=-1):}`

Notice that the single variable `y` is easily isolated in the first equation.

Taking the first equation `-7x+y=-19`, we can rewrite this by adding `7x` to both sides to see that

`color(blue)(y=7x-19)`

We now have an expression equal to `y` completely in terms of `x`. We can then take the equation we haven't used yet and replace `y` with `7x-19`, since for this system we know these are equivalent expressions.

`-2x+3color(blue)y=-1" "=>" "-2x+3color(blue)((7x-19))=-1`

We can now solve this equation, since it's entirely in terms of `x`. Distributing the `3` into the parentheses gives

`-2x+(21x-57)=-1`

Combining the `x` terms then adding `57` to both sides gives

`19x-57=-1`

`19x=56`

Then

`color(red)(x=56/19`

Now we can plug this into either equation to find the value of `y`:

`-2color(red)x+3y=-1`

`-2color(red)((56/19))+3y=-1`

`-112/19+3y=-1`

Multiplying everything by `19` yields

`-112+57y=-19`

So

`57y=93`

`y=93/57`

Both of these are divisible by `3`:

`y=(31xx3)/(19xx3)`

`color(green)(y=31/19)`

So the solution is the point `(color(red)x,color(green)y)=(color(red)(56/19),color(green)(31/19))`.

`" "`

The graphs of the the lines `-7x+y=-19` and `-2x+3y=-1` should intersect at the point `(56/19,31/19)approx(2.95,1.63)`:

graph{(-7x+y+19)(-2x+3y+1)=0 [-10.61, 17.87, -4.8, 9.44]}

They do!