Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `12x + 4y = -4` and `2x -y =6`?

Answer 1

See a solution process below:

Explanation:

To graph each line we need to find two points of solution for the equation, map the points then draw a line through the two points:

Equation 1

For `x = -2`

`(12 xx -2) + 4y = -4`

`-24 + 4y = -4`

`color(red)(24) - 24 + 4y = color(red)(24) - 4`

`0 + 4y = 20`

`4y = 20`

`(4y)/color(red)(4) = 20/color(red)(4)`

`y = 5` or `(-2, 5)`

For `x = 0`

`(12 xx 0) + 4y = -4`

`0 + 4y = -4`

`4y = -4`

`(4y)/color(red)(4) = -4/color(red)(4)`

`y = -1` or `(0, -1)`

graph{((x+2)^2+(y-5)^2-0.05)(x^2+(y+1)^2-0.05)(12x+4y+4)=0 [-15, 15, -7.5, 7.5]}

Equation 2

For `x = 0`

`(2 xx 0) - y = 6`

`0 - y = 6`

`-y = 6`

`color(red)(-1) xx -y = color(red)(-1) xx 6`

`y = -6` or `(0, -6)`

For `y = 0`

`2x - 0 = 6`

`2x = 6`

`(2x)/color(red)(2) = 6/color(red)(2)`

`x = 3` or `(3, 0)`

graph{(2x-y-6)((x-3)^2+y^2-0.05)(x^2+(y+6)^2-0.05)(12x+4y+4)=0 [-15, 15, -7.5, 7.5]}

We can see the lines intersect at `(1, -4)` for the solution to the problem.

graph{(2x-y-6)((x-1)^2+(y+4)^2-0.0125)(12x+4y+4)=0 [-8, 8, -7, 1]}

Because there is at least one solution the system of equations is consistent.