Question

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Write an equation with a variable on both sides of the equal sign that has infinitely many solutions. Solve the equation and explain why it has an infinite number of solutions.

Answer 1

One example of an equation with a variable on both sides of the equal sign that has infinitely many solutions is x - 2 = x - 2.

To solve the equation:

Start with the given equation: x - 2 = x - 2
Check that both sides of the equation are equal:
x - 2 = x - 2
The equation is already true for any value of x, so it has infinitely many solutions.
This equation has an infinite number of solutions because both sides of the equation are equal for any value of x. No matter what value x takes, the equation will always be true. This is because the left side of the equation is always the same as the right side of the equation.

Another example of equation would be 2x = 2x, which has infinite solutions.

Hence, any equation with a variable on both sides of the equal sign with the same value is true for any value of that variable and is referred to as infinitely many solutions.

Answer 2

The equation has to be a true statement.

Start with a true statement: 8 = 8

Now add 4x to both sides: 4x + 8 = 4x + 8
It's still a true statement.

Factor the right side: 4x + 8 = 4 (x+2)

Subtract 3 from both sides: 4x + 5 = 4(x+2) - 3

There's an example of an equation with infinitely many solutions. To solve it, work the process we used to create it in reverse.