Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `6x - 7y = 49` and `7y - 6x = -49`?

Answer 1

See a solution process below:

Explanation:

First, we need to graph each line by finding two solutions for each equation, plotting the solution points and then drawing a line through the points.

Equation 1:

• First Point: `x = 0`

`(6 xx 0) - 7y = 49`

`0 - 7y = 49`

`(-7y)/color(red)(-7) = 49/color(red)(-7)`

`y = -7` or `(0, -7)`

• Second Point: `x = 7`

`(6 xx 7) - 7y = 49`

`42 - 7y = 49`

`42 - color(red)(42) - 7y = 49 - color(red)(42)`

`0 - 7y = 7`

`(-7y)/color(red)(-7) = 7/color(red)(-7)`

`y = -1` or `(7, -1)`

-First Graph:

graph{(x^2 + (y+7)^2 - 0.075)((x - 7)^2 + (y+1)^2 - 0.075)(6x - 7y - 49) = 0 [-20, 20, -10, 10]}

Equation 2:

• First Point: `x = 0`

`7y - (6 xx 0) = -49`

`7y - 0 = -49`

`(7y)/color(red)(7) = -49/color(red)(7)`

`y = -7` or `(0, -7)`

• Second Point: `x = 7`

`7y - (6 xx 7) = -49`

`7y - 42 = -49`

`7y - 42 + color(red)(42) = -49 + color(red)(42)`

`7y - 0 = -7`

`(7y)/color(red)(7) = -7/color(red)(7)`

`y = -1` or `(7, -1)`

-First and Second Graph:

graph{(x^2 + (y+7)^2 - 0.075)((x - 7)^2 + (y+1)^2 - 0.075)(6x - 7y - 49) = 0 [-20, 20, -10, 10]}

Solutoin:

As we can see from the graph, both equations represent the same line. Therefore, there are an infinite number of solutions.

The lines are Consistent because there is at least one solution.