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 equations in a system by multiplication to create opposite coefficients for one variable. Add or subtract linear equations to eliminate one variable and solve for the remaining variable. Substitute a solved variable's value back into an original equation to find the value of the second variable. Identify systems with no solution (inconsistent) or infinitely many solutions (dependent) when using the elimination method. Verify the solution by substituting the ordered pair into both original equations. Set up and solve a system of linear equations from a real-world word problem using elimination. Imagine you're an e...

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 2x + y = 5 and 3x - y = 10 form a system. The solution is the ordered pair (x, y) that makes both statements true. Elimination MethodAn algebraic technique for solving a system of equations by adding or subtracting the equations to 'eliminate' one of the variables.Given x + y = 10 and x - y = 2, adding the two equations eliminates 'y', resulting in 2x = 12. CoefficientThe numerical factor multiplied by a variable in an algebraic term.In the term -7y, the coefficient is -7. Opposite CoefficientsCoefficients of the same variable in two differe...

Addition Property of Equality If A = B and C = D, then A + C = B + D This is the fundamental principle behind the elimination method. It allows us to add two equations together, combining the left sides and the right sides, to create a new, valid equation. Multiplication Property of Equality If A = B, then cA = cB for any non-zero constant c This rule allows us to multiply an entire equation by a constant to create a new, equivalent equation. We use this to create opposite coefficients for a variable if they don't already exist.