Solving Systems

Front

What is a system of equations?

Back

A system of equations is a set of two or more equations with the same variables. The solution to the system is the set of values that satisfy all equations simultaneously.

Front

What does it mean if a system of equations has infinitely many solutions?

Back

It means that there are an infinite number of points that satisfy all equations in the system, often occurring when the equations represent the same line.

Front

What is a unique solution in a system of equations?

Back

A unique solution occurs when there is exactly one set of values for the variables that satisfies all equations in the system.

Front

What does it mean if a system of equations has no solution?

Back

It means that there are no values for the variables that can satisfy all equations in the system, often occurring when the equations represent parallel lines.

Front

How can you determine if a system of equations has one, none, or infinitely many solutions?

Back

You can determine this by analyzing the slopes and intercepts of the lines represented by the equations. If the lines intersect at one point, there is one solution; if they are parallel, there is no solution; if they are the same line, there are infinitely many solutions.

Front

What is the graphical representation of a system of equations?

Back

The graphical representation of a system of equations is the intersection of the lines or curves represented by the equations on a coordinate plane.

Front

What is the substitution method in solving systems of equations?

Back

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation to find the values of the variables.

Front

What is the elimination method in solving systems of equations?

Back

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable.

Front

What is the significance of the coefficients in a system of equations?

Back

The coefficients determine the slope and position of the lines represented by the equations, affecting whether they intersect, are parallel, or are the same line.

Front

What does it mean for two lines to be parallel in the context of systems of equations?

Back

Two lines are parallel if they have the same slope but different y-intercepts, indicating that they will never intersect and thus have no solution.