Question

Make two equations for each:

Answer 1

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