Example Equation With One Solution: 7x+2x-3=18+2x Example Equation With No Solution: x + 13=x + 2 Example Equation With Infinitely Many Solutions: 12x +12 = 12x +17 - 5
Practice makes perfect
We want to write three equations — one that has one solution, one that has no solution, and one with infinitely many solutions. Let's write them one at a time!
Equation With One Solution
We can write any equation in reverse order to solving it. We start from the solution and then we use Properties of Equality to write an equation with more terms. Our equation will have one solution, which means that one variable will have one value. We will arbitrarily choose a variable x and a value 3.
x=3
We will use Properties of Equality to apply the same operations on both sides of the equation. We will get an equivalent equation with the same solution but with more terms. Let's start with multiplication. We can multiply both sides of our equation by any number other than zero, let's say 7.
We got an example equation with exactly one solution. Additionally, we know that this solution is 3.
Equation With No Solution
Next, we want to write an equation with no solution. Let's write an equation with one variable.
x=x
This equation is always true, because x is always equal to x. Next, let's add two different numbers to two sides of the equation.
x + 13=x + 2
We can see that our equation is no longer true. After adding two different numbers to both sides, this equation is always false. This way we got an equation with no solution.
Equation With Infinitely Many Solutions
Finally, we will write an equation with infinitely many solutions. We know that an equation has infinitely many solutions when it is always true. In other words, when after simplifications both sides of the equation are equal for all values of x.
x=x
This equation is always true, because x is always equal to x. In a similar process as before, we can use Properties of Equality to apply the same operations on both sides of the equation. Let's start with multiplication.
We got an equation equivalent to x=x, so it has infinitely many solutions. Notice, that all the equations we wrote are just example equations. There are infinitely many different equations with one, none, or infinitely many solutions.
