Math  /  Algebra

QuestionHow do you know if an equation has one solution, no solution, or infinitely many solutions?
An equation that has one solution simplifies to \square an equation that has no solution simplifies \square and an equation that has fies to \square a true statement, x=0,\mathrm{x}=0, x=ax=a number, a false statement,

Studdy Solution

STEP 1

What is this asking? How can we tell if an equation has one, zero, or infinite solutions? Watch out! Don't mix up "no solution" with "x=0\mathrm{x}=0".
A solution of x=0\mathrm{x}=0 *is* a solution!
It means zero is the one and only number that makes the equation true.

STEP 2

1. One Solution
2. No Solution
3. Infinite Solutions

STEP 3

When an equation has *one* solution, it means there's only *one* specific value for the variable that makes the equation true!
Think of it like a lock and key: only *one* key fits.

STEP 4

Mathematically, after simplifying, we get the variable equal to a specific number.
For example, x=5\mathrm{x} = 5.
Here, the *one* solution is x=5\mathrm{x} = 5.

STEP 5

Imagine trying to fit a square peg into a round hole.
It just won't work!
Similarly, an equation with *no* solution means there's *no* value for the variable that will ever make the equation true.

STEP 6

When we simplify such an equation, we end up with something nonsensical, like 2=52 = 5.
This is a **false statement**, and it tells us there's *no* solution.

STEP 7

Picture a coin.
Two sides, same coin!
An equation with infinite solutions means any value you plug in for the variable will make the equation true.

STEP 8

After simplifying, we'll get a **true statement** where both sides of the equation are equal, like 3=33 = 3 or x=x\mathrm{x} = \mathrm{x}.
This means any value for x\mathrm{x} works!

STEP 9

An equation with one solution simplifies to x=a number\mathrm{x} = \text{a number}.
An equation with no solution simplifies to a false statement.
An equation with infinitely many solutions simplifies to a true statement.

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