Answer:
Two equations with exactly one solution
y=x
y=-x
Two equations with no solutions
y=x
y=x+1
Two equations with infinitely many solutions
x+y+1
2x+2y=2
Step-by-step explanation:
Let's use linear equations. The slope intercept form is
y=mx+b
m is the slope and b is the y intercept.
To make the equations have one solution, the two equations must represent lines that do not have the same slope, so that they intersect. We must choose different slopes, but any y intercepts will do. Example: Choose the slopes to be 1 and -1 and choose both y intercepts to be 0
y=x
y=-x
For there to be no solutions, all we need are the same slopes and different y intercepts, so that the lines are parallel and at different heights. Let's choose the slopes to be 1 and the y intercepts to be 0 and 1.
y=x
y=x+1
To get infinitely many solutions all we need is for the lines to be parallel with the same y intercept, so that they lie on top of each other. We could use
y=x
y=x
but your teacher probably wants something different. So, lets use another form of a linear equation, the so-called standard form.
ax+by=c
if you solve this for y, you get
y=-(a/b)x+(c/b)
This is in the y=mx+b form we used before, except that now the slope is -(a/b) and the y intercept is (c/b). So, we will get the same slope if the ratio of a to b is the same for both equations and the ratio of c to b is the same for both equations. In other words, simply choose a standard form equation and then get another by multiplying through by any number. In general
ax+by=c
and
kax+kby=kc
will do the trick. Here is an example.
x+y+1
2x+2y=2
