Find the number of solutions to a system of equations by graphing

Learning Objectives Define a system of linear equations and its solution. Accurately graph two linear equations on the same coordinate plane. Identify the intersection point of two lines, if it exists. Determine if two lines are parallel or coincident from their graphs. Classify a system of linear equations as having one solution, no solution, or infinitely many solutions based on its graph. Explain the relationship between the graphical representation of lines and the number of solutions to a system. Ever wonder if two paths will ever cross? 🗺️ Or if they'll always run side-by-side, never meeting? In this lesson, you'll learn how to use graphs to visually determine if two linear equations have one solution, no solutions, or many solutions. This skill helps us und...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that use the same variables.Equation 1: $y = 2x + 1$ Equation 2: $y = -x + 4$ Solution to a SystemA set of values for the variables that makes ALL equations in the system true. Graphically, it's the point(s) where the lines intersect.For the system $y=x+1$ and $y=2x-1$, the point $(2,3)$ is a solution because $3=2+1$ and $3=2(2)-1$ are both true. Intersecting LinesTwo lines that cross each other at exactly one point.The lines $y=x$ and $y=-x+2$ intersect at $(1,1)$. Parallel LinesTwo lines that are always the same distance apart and never intersect. They have the same slope but different y-intercepts.The lines $y=3x+2$ and $y=3x-1$ are parallel. Coincident LinesTwo lines that lie exactly on top of e...

Rule for One Solution If the graphs of two linear equations intersect at exactly one point, the system has one unique solution. This occurs when the lines have different slopes. The point of intersection $(x,y)$ is the single solution to the system. Rule for No Solution If the graphs of two linear equations are parallel and never intersect, the system has no solution. This occurs when the lines have the same slope but different y-intercepts. Since they never cross, there is no point that satisfies both equations simultaneously. Rule for Infinitely Many Solutions If the graphs of two linear equations are coincident (the exact same line), the system has infinitely many solutions. This occurs when the lines have the same slope and the same y-intercept. Every point on th...