Solve multi-variable equations

Learning Objectives Isolate a specific variable in a complex literal equation involving exponents, roots, and fractions. Solve systems of two linear equations with two variables using both the substitution and elimination methods. Solve systems of three linear equations with three variables using systematic elimination. Solve non-linear systems of equations, specifically those involving a line and a conic section (e.g., a parabola or circle). Interpret the results of a system to determine if it has one solution, no solution, or infinitely many solutions. Translate a real-world word problem into a system of multi-variable equations and solve it. Ever wondered how GPS pinpoints your exact location using signals from multiple satellites? 🛰️ It's all done by solving a syste...

TermDefinitionExample Multi-variable Equation (Literal Equation)An equation that contains two or more variables. Often, the goal is to express one variable in terms of the others.The formula for the area of a trapezoid, A = (1/2)h(b_1 + b_2), is a multi-variable equation with variables A, h, b_1, and b_2. System of EquationsA collection of two or more equations that share the same set of variables.The set of equations { y = 2x + 1, y = -x + 4 } is a system of two equations with two variables, x and y. Solution to a SystemA set of values for the variables that makes all equations in the system true at the same time. For a 2D system, this is the point of intersection of the graphs.For the system { y = 2x + 1, y = -x + 4 }, the solution is (x=1, y=3) because 3 = 2(1) + 1 and 3 = -(1) + 4 are...

The Substitution Method 1. Solve one equation for one variable. 2. Substitute this expression into the other equation. 3. Solve the resulting equation. 4. Back-substitute to find the other variable(s). This method is most efficient when one of the equations can be easily solved for a variable (e.g., y = ... or x = ...). The Elimination Method 1. Align equations by variable. 2. Multiply one or both equations by constants so that the coefficients of one variable are opposites. 3. Add the equations together to eliminate that variable. 4. Solve for the remaining variable and back-substitute. This method is ideal when variables are already aligned in standard form (Ax + By = C) and coefficients are the same, opposite, or easy multiples of each other.