Solve systems of linear equations

Learning Objectives Identify if a system of linear equations has one solution, no solution, or infinitely many solutions by analyzing their equations. Solve systems of linear equations by graphing the lines and finding the point of intersection. Accurately solve systems of linear equations using the substitution method. Efficiently solve systems of linear equations using the elimination method, including cases that require multiplication. Determine the most appropriate method (graphing, substitution, or elimination) to solve a given system. Model and solve real-world scenarios by setting up and solving a system of linear equations. Ever compared two phone plans to find the cheapest option? 📱 You're using the same logic as solving a system of equations to find where the...

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 + 4 form a system. The goal is to find the (x, y) pair that works for both. Solution to a SystemAn ordered pair (x, y) that makes all equations in the system true. Geometrically, it is the point where the lines intersect.For the system y = x + 2 and y = 3x - 2, the solution is (2, 4) because 4 = 2 + 2 and 4 = 3(2) - 2 are both true. Consistent SystemA system of equations that has at least one solution. The lines intersect at one point or are the exact same line.The system y = 2x and y = x + 1 is consistent because it has one solution at (1, 2). Inconsistent SystemA system of equations that has no solution. Geometrically, the lines are para...

Slope-Intercept Form y = mx + b This form is essential for the graphing method and for quickly comparing two lines. 'm' represents the slope and 'b' represents the y-intercept. If two lines have different slopes, they will intersect exactly once. Conditions for Number of Solutions Given two lines y = m_1x + b_1 and y = m_2x + b_2: 1. One Solution: m_1 ≠ m_2 2. No Solution: m_1 = m_2 and b_1 ≠ b_2 3. Infinite Solutions: m_1 = m_2 and b_1 = b_2 By comparing the slopes (m) and y-intercepts (b) of the equations in a system, you can determine the number of solutions without solving.