Classify a system of equations by graphing

Learning Objectives Graph a system of two linear equations on a coordinate plane. Identify the point of intersection on a graph as the solution to the system. Classify a system as consistent and independent when its graph shows one point of intersection. Classify a system as inconsistent when its graph shows parallel lines. Classify a system as consistent and dependent when its graph shows coincident (the same) lines. Determine if a system has one solution, no solution, or infinitely many solutions based on its graph. Imagine two ships sailing on the ocean. Will their paths cross, will they travel side-by-side forever, or are they on the exact same course? 🚢 Let's find out using graphs! In this tutorial, you'll learn how to transform systems of equations into pic...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables.The equations y = 2x + 1 and y = -x + 7 form a system. Solution of a SystemAn ordered pair (x, y) that makes all equations in the system true. Graphically, it is the point where the lines intersect.For the system y = x + 2 and y = 4, the solution is (2, 4) because it lies on both lines. Consistent SystemA system of equations that has at least one solution. The graphs of the lines intersect.A system whose lines cross at a single point or are the exact same line. Inconsistent SystemA system of equations that has no solution. The graphs of the lines are parallel and never intersect.The lines y = 2x + 3 and y = 2x - 1 form an inconsistent system. Independent SystemA consistent s...

One Solution (Consistent & Independent) For two lines y = m_1x + b_1 and y = m_2x + b_2, there is one solution if m_1 \neq m_2. If the slopes of the two lines are different, the lines are guaranteed to intersect at exactly one point. No Solution (Inconsistent) For two lines y = m_1x + b_1 and y = m_2x + b_2, there is no solution if m_1 = m_2 and b_1 \neq b_2. If the slopes are the same but the y-intercepts are different, the lines are parallel and will never intersect. Infinite Solutions (Consistent & Dependent) For two lines y = m_1x + b_1 and y = m_2x + b_2, there are infinite solutions if m_1 = m_2 and b_1 = b_2. If the slopes and y-intercepts are identical, the equations represent the exact same line. Every point on the line is a solution.