Find the number of solutions to a system of equations

Learning Objectives Identify the three possible types of solutions for a system of two linear equations: one solution, no solution, and infinitely many solutions. Relate the number of solutions to the graphical representation of the lines as intersecting, parallel, or coincident. Determine the number of solutions by comparing the slopes and y-intercepts of equations in slope-intercept form (y = mx + b). Determine the number of solutions by analyzing the ratios of coefficients and constants in standard form (Ax + By = C). Interpret the result of an algebraic process (like elimination or substitution) to determine if a system has one, none, or infinite solutions. Classify a system as consistent, inconsistent, independent, or dependent based on its number of solutions. Imagine...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables.The equations y = 2x + 3 and y = -x + 9 form a system. Consistent SystemA system of equations that has at least one solution. The lines intersect at one or more points.The system y = x + 1 and y = 2x - 1 is consistent because it has one solution at (2, 3). Inconsistent SystemA system of equations that has no solution. Graphically, the lines are parallel and never intersect.The system y = 2x + 4 and y = 2x - 1 is inconsistent. The lines have the same slope but different y-intercepts. Independent SystemA consistent system that has exactly one unique solution. Graphically, the lines intersect at a single point.The system y = 3x and y = x + 2 is independent. The lines have diffe...

Analysis in Slope-Intercept Form For a system with equations y = m₁x + b₁ and y = m₂x + b₂: 1. One Solution: m₁ ≠ m₂ 2. No Solution: m₁ = m₂ and b₁ ≠ b₂ 3. Infinite Solutions: m₁ = m₂ and b₁ = b₂ Convert both equations into slope-intercept form (y = mx + b). Then, compare their slopes (m) and y-intercepts (b) to quickly determine the number of solutions without graphing. Analysis in Standard Form For a system with equations A₁x + B₁y = C₁ and A₂x + B₂y = C₂: 1. One Solution: A₁/A₂ ≠ B₁/B₂ 2. No Solution: A₁/A₂ = B₁/B₂ ≠ C₁/C₂ 3. Infinite Solutions: A₁/A₂ = B₁/B₂ = C₁/C₂ This is a powerful shortcut that works when both equations are in standard form (Ax + By = C). By comparing the ratios of the coefficients of x, the coefficients of y, and the constants, you can classify the...