Solve a system of equations by graphing

Learning Objectives Graph linear equations accurately in two variables on the Cartesian plane. Identify the point of intersection of two graphed lines as the solution to the system. Classify a system of equations as consistent (independent or dependent) or inconsistent based on its graphical representation. Solve systems of linear equations by graphing them and finding their intersection point. Verify the solution to a system by substituting the coordinates into both original equations. Recognize and interpret the graphical meaning of systems with no solution (parallel lines) and infinitely many solutions (coincident lines). Ever tried to find the exact spot where two roads cross on a map? 🗺️ Solving a system of equations by graphing is the mathematical equivalent of finding...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables.y = 2x + 3 and y = -x + 6 form a system of two linear equations with variables x and y. Solution to a SystemAn ordered pair (x, y) that satisfies every equation in the system. Graphically, this is the point where all the lines in the system intersect.For the system y = x + 2 and y = -x + 4, the solution is (1, 3) because if you substitute x=1 and y=3, both equations are true. Point of IntersectionThe specific coordinate point (x, y) where the graphs of two or more lines cross each other.The graphs of y = 2x and y = -x + 3 intersect at the point (1, 2). Consistent SystemA system of equations that has at least one solution. The graphs intersect at one point (independent) or ar...

Slope-Intercept Form y = mx + b This is the most direct form for graphing a linear equation. 'm' represents the slope (rise over run) of the line, and 'b' represents the y-intercept, which is the point (0, b) where the line crosses the vertical y-axis. Solution Verification Principle Given a system with equations f(x) and g(x), a solution (x_s, y_s) must satisfy y_s = f(x_s) AND y_s = g(x_s). To confirm that a point of intersection is the correct solution, you must substitute its x and y coordinates into *all* equations in the system. Both substitutions must result in true statements. Parallel and Coincident Lines For two lines y = m_1x + b_1 and y = m_2x + b_2: 1. If m_1 = m_2 and b_1 ≠ b_2, the lines are parallel (no solution). 2. If m_1 = m_2 a...