Question

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

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:

                      $\displaystyle{ x- \frac{3}{2} =3^x$.

Note that $y=x- \frac{3}{2}$ 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 $3^{\frac{3}{2}}\approx5.2$ and is increasing very fast.


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


Answer: no solution.