Solve a system of equations using any method

Learning Objectives Identify the most appropriate method (graphing, substitution, or elimination) to solve a given system of linear equations. Solve a system of linear equations by graphing and interpret the point of intersection as the solution. Solve a system of linear equations algebraically using the substitution method. Solve a system of linear equations algebraically using the elimination (or linear combination) method. Classify a system as consistent (independent or dependent) or inconsistent based on its solution. Translate a real-world problem into a system of linear equations and solve it. Verify the solution to a system by substituting the values back into the original equations. Ever tried to figure out the exact price of a slice of pizza and a soda when you on...

TermDefinitionExample System of Linear EquationsTwo or more linear equations that share the same variables. The goal is to find the variable values that satisfy all equations at the same time.The equations y = 2x + 1 and y = -x + 7 form a system. The solution is the point (x, y) that lies on both lines. 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 = -x + 4, the solution is (1, 3) because 3 = 1 + 2 and 3 = -1 + 4 are both true statements. 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). Inconsiste...

The Substitution Method 1. Isolate a variable in one equation (e.g., solve for y). \n2. Substitute the resulting expression into the other equation. \n3. Solve the new equation for the remaining variable. \n4. Back-substitute to find the first variable. This method is most efficient when one of the variables in one of the equations already has a coefficient of 1 or -1, making it easy to isolate. The Elimination Method 1. Write both equations in standard form (Ax + By = C). \n2. Multiply one or both equations by a constant to make the coefficients of one variable opposites (e.g., 3x and -3x). \n3. Add the revised equations together to eliminate one variable. \n4. Solve for the remaining variable and back-substitute. This method is most efficient when the equations are already...