Make two equations for each: exactly one solution
no solutions
infinitely many solutions

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

proofs involving isosceles triangles often require special consideration because an isosceles triangle has several distinct properties that do not apply to normal triangles.(more about triangle types) therefore, when you are trying to prove that two triangles are congruent, and one or both triangles, are isosceles you have a few theorems that you can use to make your life easier.

step-by-step explanation:

an isosceles triangle has two congruent sides and two congruent angles. the congruent angles are called the base angles and the other angle is known as the vertex angle. ∠bac and ∠bca are the base angles of the triangle picture on the left. the vertex angle is ∠abc