One Solution, No Solution, or Infinite Solutions

Front

What does it mean when an equation has no solution?

Back

An equation has no solution when there are no values for the variable that can satisfy the equation, often resulting in a contradiction.

Front

What is an example of an equation with no solution?

Back

An example is 2x + 1 = 2x - 1, which simplifies to 1 = -1, a contradiction.

Front

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

Back

An equation has infinitely many solutions when any value for the variable satisfies the equation, often resulting in an identity.

Front

What is an example of an equation with infinitely many solutions?

Back

An example is 3x + 15 = 3(x + 5), which simplifies to 3x + 15 = 3x + 15, an identity.

Front

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

Back

By simplifying the equation and checking for contradictions (no solution) or identities (infinitely many solutions).

Front

What is a contradiction in the context of equations?

Back

A contradiction occurs when the simplification of an equation leads to a false statement, such as 1 = -1.

Front

What is an identity in the context of equations?

Back

An identity is a statement that is always true, such as 3x + 15 = 3x + 15.

Front

What is the significance of the coefficients in a linear equation?

Back

The coefficients determine the slope and position of the line represented by the equation, affecting the number of solutions.

Front

What is the graphical representation of an equation with no solution?

Back

The graphical representation is two parallel lines that never intersect.

Front

What is the graphical representation of an equation with infinitely many solutions?

Back

The graphical representation is a single line, where every point on the line is a solution.