a To have one solution of an equation, you have to be able to isolate x.
B
b To have no solution of an equation, you have to end up with a contradiction.
C
c If the equation does not have only one solution but it does have a solution, what is the remaining option?
A
aExample Solution: a=-1, b=1, c=2, and d=0
B
bExample Solution: a=3, b=4, c=3, and d=5
C
cExample Solution: a=6, b=1, c=6, and d=1
Practice makes perfect
a Student A claims that the equation will always have one solution. To support this claim we have to select values for a, b, c, and d such that x does not cancel out of the equation when simplifying. We only need that a and c are different numbers. We will arbitrarily choose -1 and 2.
a= -1 and c= 2
Now we can choose any values for b and d.
b= 1 and d= 0
Next, we substitute those values in our equation.
-1x+ 1= 2x+ 0 ⇓
- x+1=2x
Let's solve this equation.
There is one solution to this equation. Our selection made Student A's claim correct. Note that this is an example solution. As long as we choose different values for a and c we will get an equation with only one solution.
b Student B claims that the equation will always have no solution. This time we have to select values for a, b, c, and d such that x cancels out of the equation and leaves us with a contradiction. This means a and c have to be the same numbers. We will arbitrarily choose 3.
a=c= 3
After x cancels out, we are left with the equation b=d. Since we want a contradiction, those numbers have to be different.
b= 4 and d= 5
Let's substitute those values in our equation.
3x+ 4= 3x+ 5
Let's try to solve this equation.
Since we are left with a contradiction this equation has no solution, supporting Student B's claim. Note that this is an example solution. As long as we choose the same values for a and c while choosing different values for b and d we will get an equation with no solution.
c Finally, we want both students to be incorrect. This means that we want the equation to have infinitely many solutions — we want it to be an identity. To create an identity, a and c should be equal, and b and d should also be equal. We will arbitrarily choose 6 and 1.
a=c= 6 and b=d= 1Once more, we will substitute these values into our equation.
6x+ 1= 6x+ 1
Finally, we will try to solve this equation.
Since we got an identity, our equation has infinitely many solutions. Now both students are incorrect. Note that this is an example solution. As long as we choose the same values for a and c and also choose the same values for b and d we will get an equation with infinitely many solutions.
