Question

What is the solution to this system of equations? `{:(y=10+6x),(y=6x):}`

Answer 1

There is no solution.

Explanation:

These two lines have the same slope, that being 6. When two lines have the same slope, they'll either be overlapping (i.e. the same line) or distinctly parallel.

When two lines overlap, they share all points in common. That is, any point `(x,y)` that lies on one line will also lie on the other line. (We say there are infinitely many solutions.)

When two lines are distinct and parallel, they share no points in common, because parallel lines never cross. That is, no point `(x,y)` that falls on one line can ever fall on the other line. (We say there are no solutions.)

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To show this mathematically, we start with `y=10+6x` and, since we're trying to see if this equation is "compatible" with `y=6x`, we replace `y` with `6x` to see if we can find (at least) one `x`-value that works for both lines.

`y"            "=10+6x`

`6x"          "=10+6x" "`(assume `y` is equal to `6x`)

`6x-6x=10`

`"            "0=10`

This "equality" gives no solution in `x`; there is no `x`-value that could ever make `0` equal to `10.` That means our assumption `y=6x` is not compatible with the first equation `y=10+6x`. Thus, there is no `(x,y)` point that satisfies both equations.