Question

How do you solve the system of equations `y= 6x + 5` and `y = 6x + 2`?

Answer 1

See a solution process below:

Explanation:

Both equations are in the slope-intercept form. The slope-intercept form of a linear equation is: `y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b)` is the y-intercept value.

`y = color(red)(6)x + color(blue)(5)`

`y = color(red)(6)x + color(blue)(2)`

Both equations have the same slope: `color(red)(m = 6)`

Because both equations have the same slope they are, by definition, either the same line or parallel lines.

However, both equations have a different `y`-intercept.

The first equations `y`-intercept is: `color(blue)(b = 5)`

The second equations `y`-intercept is: `color(blue)(b = 2)`

If both equations represented the same line they would have each and every point in common. However, because they have different `y`-intercepts they must represent different but parallel lines.

Therefore, the is no solution to this system of equations. Or the solution is the null or empty set: `{O/}`