Front

What are parallel lines in the context of linear equations?

Back

Parallel lines are lines in a plane that never intersect. They have the same slope but different y-intercepts.

Front

What does it mean for a solution to a system of equations?

Back

A solution to a system of equations is an ordered pair (x, y) that satisfies all equations in the system, making them true.

Front

How many solutions can a system of linear equations have?

Back

A system of linear equations can have one solution (intersecting lines), no solutions (parallel lines), or infinitely many solutions (coincident lines).

Front

What is the graphical representation of a system with no solutions?

Back

The graphical representation of a system with no solutions consists of two parallel lines that never intersect.

Front

How do you solve a system of equations using substitution?

Back

To solve using substitution, solve one equation for one variable and substitute that expression into the other equation.

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 other.

Front

What does it mean to eliminate a variable in a system of equations?

Back

To eliminate a variable means to manipulate the equations so that one variable is removed, allowing for easier solving of the remaining variable.

Front

What is the significance of the slopes of two lines in a system of equations?

Back

The slopes determine the relationship between the lines: if they are equal, the lines are parallel (no solutions); if they are different, they intersect (one solution); if they are the same and the y-intercepts are also the same, they are coincident (infinitely many solutions).

Front

How do you determine the number of solutions in a system of equations?

Back

You can determine the number of solutions by analyzing the slopes and y-intercepts of the lines represented by the equations.

Front

What is the first step in solving the system 3x - y = 7 and 2x + y = 3?

Back

The first step can be to use either substitution or elimination to solve for one variable.