Classifying Solutions to Linear Systems of Equations

Front

What is a linear equation?

Back

A linear equation is an equation that makes a straight line when graphed. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What does it mean for two lines to intersect?

Back

Two lines intersect when they cross each other at a single point, which represents the unique solution to the system of equations.

Front

What are parallel lines?

Back

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

Front

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

Back

If a system of equations has no solution, it means the lines are parallel and will never intersect.

Front

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

Back

If a system of equations has infinitely many solutions, it means the lines coincide, or are the same line, and every point on the line is a solution.

Front

How can 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. If the slopes are different, there is one solution; if they are the same but y-intercepts differ, there is no solution; if they are the same and y-intercepts are the same, there are infinitely many solutions.

Front

What is the slope of a line?

Back

The slope of a line is a measure of its steepness, usually represented by 'm' in the equation y = mx + b. It is calculated as the change in y divided by the change in x.

Front

What is a y-intercept?

Back

The y-intercept is the point where a line crosses the y-axis, represented by 'b' in the equation y = mx + b.

Front

Given the equations y = 2x + 3 and y = 2x - 1, how many solutions does this system have?

Back

This system has no solution because the lines are parallel (same slope, different y-intercepts).

Front

What is the solution to the system of equations y = -x + 5 and y = 2x - 7?

Back

The solution is (4, 1), where the two lines intersect.