Solve a system of equations using elimination

Learning Objectives Identify when the elimination method is the most efficient strategy for solving a system of linear equations. Manipulate one or both linear equations by multiplication to create opposite coefficients for a chosen variable. Add or subtract linear equations vertically to eliminate one variable. Solve for the remaining variable and use substitution to find the value of the eliminated variable. State the solution as an ordered pair (x, y). Verify the solution by substituting the ordered pair into both original equations. Recognize systems that result in no solution or infinitely many solutions. You and a friend go to a cafe. You buy 2 coffees and 1 pastry for $8, while your friend buys 2 coffees and 3 pastries for $14. How can you figure out the exact price...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables. The goal is to find the values of the variables that satisfy all equations in the set simultaneously.The equations 3x + 2y = 7 and x - y = 4 form a system of linear equations. Solution to a SystemAn ordered pair (x, y) that makes every equation in the system a true statement. Geometrically, it is the point where the lines represented by the equations intersect.For the system x + y = 5 and x - y = 1, the solution is (3, 2) because 3 + 2 = 5 and 3 - 2 = 1. Elimination MethodAn algebraic method for solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.Adding the equations x + y = 5 and x - y = 1 results in 2x = 6, which e...

Addition Property of Equality If A = B and C = D, then A + C = B + D This is the fundamental principle behind elimination. If two equations are true, you can add the left sides together and the right sides together, and the resulting equation will also be true. This allows us to combine the equations in a system. Multiplication Property of Equality If A = B, then cA = cB for any non-zero constant c This rule allows us to create opposite coefficients when they don't already exist. We can multiply every term in an equation by a chosen number to create an equivalent equation that is more useful for elimination.