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 a chosen variable. Add or subtract equations vertically to eliminate one variable and solve for the other. Use back-substitution to find the value of the eliminated variable. Solve systems of three linear equations with three variables using the elimination method. Set up and solve a system of linear equations to determine the coefficients of a polynomial function given specific points. Interpret the solution as the intersection point of linear functions or as parameters of a more complex function. How can we find the exact coefficients of a quadra...

TermDefinitionExample System of Linear EquationsA set of two or more linear equations that share the same variables. The solution to the system is the ordered pair (or triple, etc.) that satisfies all equations simultaneously.The functions `f(x) = 2x - 1` and `g(x) = -x + 5` can be written as a system: `y = 2x - 1` and `y = -x + 5`. The solution `(2, 3)` is the point of intersection where `f(2) = g(2)`. Elimination MethodAn algebraic method for solving a system of equations where you add or subtract the equations to eliminate one of the variables.For the system `x + y = 10` and `x - y = 4`, adding the two equations eliminates `y`, resulting in `2x = 14`. CoefficientThe numerical factor of a term. In the elimination method, we manipulate equations to make the coefficients of one variable o...

Addition Property of Equality If `A = B` and `C = D`, then `A + C = B + D`. This property is the foundation of the elimination method. It allows us to add two equations together. Since each equation represents two equal quantities, adding them results in a new, valid equation. Multiplication Property of Equality If `A = B`, then `kA = kB` for any non-zero constant `k`. This property allows us to multiply an entire equation by a constant without changing its solution set. We use this to create opposite coefficients before adding the equations.