Question

In two or more complete sentences, give the solution and explain how you solved the system.

y=-2x+3
y=-2*3^x

Answer 1

The system is:

i) y=-2x+3
ii)
`y=-2\cdot3^x`.

Making y's equal, we have:


`-2x+3=-2\cdot3^x`.

Dividing all terms by -2, we have:


`\displaystyle{ x- (3)/(2) =3^x`.

Note that
`y=x- (3)/(2)` is an increasing linear function. (Its graph is the same as y=x, only shifted 3/2 units down.)

The graph of this function is below the x-axis up to 3/2, where it cuts the x-axis, and then increases above the x-axis making an angle of 45° with the x-axis.

The graph of
`y=3^x` is completely above the x-axis, and it rises much faster than the line. We could compare at 3/2, for example, where the linear function is 0. The exponential function takes the value
`3^{(3)/(2)}\approx5.2` and is increasing very fast.


Thus, the graphs never meet, which means the system has no solution.


Answer: no solution.