Answer:
Example system with no solution
We're asked to find the number of solutions to this system of equations:
\begin{aligned} y &= -3x+9\\\\ y &= -3x-7 \end{aligned}
y
y
=−3x+9
=−3x−7
Without graphing these equations, we can observe that they both have a slope of -3−3minus, 3. This means that the lines must be parallel. And since the yyy-intercepts are different, we know the lines are not on top of each other.
There is no solution to this system of equations.
Example system with infinite solutions
We're asked to find the number of solutions to this system of equations:
\begin{aligned} -6x+4y &= 2\\\\ 3x-2y &= -1 \end{aligned}
−6x+4y
3x−2y
=2
=−1
Interestingly, if we multiply the second equation by -2−2minus, 2, we get the first equation:
\begin{aligned} 3x-2y &= -1\\\\ \blueD{-2}(3x-2y)&=\blueD{-2}(-1)\\\\ -6x+4y &= 2 \end{aligned}
3x−2y
−2(3x−2y)
−6x+4y
=−1
=−2(−1)
=2
In other words, the equations are equivalent and share the same graph. Any solution that works for one equation will also work for the other equation, so there are infinite solutions to the system.
Explanation: