One Solution, No Solution, or Infinite Solutions

Front

What does it mean when an equation has one solution?

Back

An equation has one solution when there is exactly one value of the variable that satisfies the equation.

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What does it mean when an equation has no solution?

Back

An equation has no solution when there is no value of the variable that can satisfy the equation, often resulting in a contradiction.

Front

What does it mean when an equation has infinitely many solutions?

Back

An equation has infinitely many solutions when any value of the variable satisfies the equation, typically when both sides of the equation are equivalent.

Front

How can you determine if an equation has one solution, no solution, or infinitely many solutions?

Back

You can determine this by simplifying the equation and analyzing the resulting expressions for contradictions or identities.

Front

What is the significance of equivalent equations in solving for variables?

Back

Equivalent equations have the same solution set, meaning that manipulating an equation to create an equivalent form can help find solutions.

Front

What is the first step in solving a linear equation?

Back

The first step is to simplify both sides of the equation, if possible, by distributing and combining like terms.

Front

What is the role of the distributive property in solving equations?

Back

The distributive property allows you to eliminate parentheses by multiplying a term outside the parentheses by each term inside.

Front

What does it mean for two expressions to be equivalent?

Back

Two expressions are equivalent if they yield the same value for all values of the variable.

Front

What is a contradiction in the context of equations?

Back

A contradiction occurs when simplifying an equation leads to a false statement, indicating that there are no solutions.

Front

What is an identity in the context of equations?

Back

An identity is an equation that is true for all values of the variable, indicating that there are infinitely many solutions.