Solve a system of equations using elimination

Learning Objectives Identify a system of linear equations. Recognize when coefficients are additive inverses (opposites). Apply the elimination method to solve systems where one variable's coefficients are already opposites. Transform equations by multiplication to create opposite coefficients for a variable. Solve systems of equations using elimination when multiplication is required for one or both equations. Verify the solution of a system of equations by substituting values back into the original equations. Interpret the solution of a system of equations as the point of intersection on a graph. Ever wondered how stores figure out how many of each item to order, or how scientists balance chemical reactions? 🤔 It often involves solving puzzles with multiple unknown...

TermDefinitionExample System of Linear EquationsTwo or more linear equations with the same variables that are considered together.The pair of equations: `x + y = 5` and `2x - y = 1`. VariableA symbol, typically a letter, used to represent an unknown numerical value in an equation.In the equation `3x + 2y = 10`, `x` and `y` are variables. CoefficientThe numerical factor that multiplies a variable in an algebraic term.In the term `5x`, `5` is the coefficient of `x`. Elimination MethodA technique for solving systems of linear equations by adding or subtracting the equations to cancel out (eliminate) one of the variables.Adding `x + y = 5` and `2x - y = 1` to eliminate `y`. Additive Inverse (Opposites)Two numbers that have the same absolute value but opposite signs, so their sum is zero.`7` a...

The Elimination Principle (Adding Equations) If $A = B$ and $C = D$, then $A + C = B + D$. This rule states that you can add two equations together (left side to left side, right side to right side) to form a new valid equation. If one variable has coefficients that are additive inverses, this addition will eliminate that variable, leaving a simpler equation with only one variable. Multiplying an Equation by a Constant If $Ax + By = C$, then $k(Ax + By) = k(C)$ for any non-zero constant $k$. This means $kAx + kBy = kC$. To create additive inverse coefficients for a variable, you can multiply one or both equations by a carefully chosen non-zero constant. It is crucial to multiply *every* term on *both* sides of the equation to maintain equality and create an equivalent equati...