Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `-2x + 4y=8` and `6x + 3y= -9`?

Answer 1

See a solution process below:

Explanation:

We can find two points for each equation, plot the two points and draw a straight line through the two points.

For Equation: `-2x + 4y = 8`

• First Point: For `x = 0`

`(-2 * 0) + 4y = 8`

`0 + 4y = 8`

`4y = 8`

`(4y)/color(red)(4) = 8/color(red)(4)`

`y = 2` or `(0, 2)`

Second Point: For `y = 0`

`-2x + (4 * 0) = 8`

`-2x + 0 = 8`

`-2x = 8`

`(-2x)/color(red)(-2) = 8/color(red)(-2)`

`x = -4` or `(-4, 0)`

We can next plot the two points on the coordinate plane and draw the straight line through them for the first equation:

graph{(4y - 2x - 8)(x^2+(y-2)^2-0.025)((x+4)^2+y^2-0.025)=0 [-10, 10, -5, 5]}

For Equation: `6x + 3y = -9`

• First Point: For `x = 0`

`(6 * 0) + 3y = -9`

`0 + 3y = -9`

`3y = -9`

`(3y)/color(red)(3) = -9/color(red)(3)`

`y = -3` or `(0, -3)`

Second Point: For `y = 3`

`6x + (3 * 3) = -9`

`6x + 9 = -9`

`6x + 9 - color(red)(9) = -9 - color(red)(9)`

`6x - 0 = -18`

`6x = -18`

`(6x)/color(red)(6) = -18/color(red)(6)`

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

We can next plot the two points on the coordinate plane and draw the straight line through them for the first equation:

graph{(6x + 3y + 9)(4y - 2x - 8)(x^2+(y+3)^2-0.025)((x+3)^2+(y-3)^2-0.025)=0 [-10, 10, -5, 5]}

We can see these two lines intersect at point: `(-2, 1)`

graph{(6x + 3y + 9)(4y - 2x - 8)((x+2)^2+(y-1)^2-0.015)=0 [-10, 2, -1, 5]}

Because there is at least one solution to this system of equations it is by definition consistent.