Solve a system of equations using substitution: word problems

Learning Objectives Translate complex word problems into a system of two equations involving linear, quadratic, or other functions. Identify the most efficient variable to isolate in a given system of equations. Accurately apply the substitution method to solve for the point of intersection between two functions. Solve the resulting single-variable equation, including quadratic equations that may arise. Back-substitute the found value to determine the value of the second variable. Interpret the numerical solution within the real-world context of the original word problem. When does a company's rising revenue finally overtake its costs to become profitable? 📈 The answer lies at the intersection of two functions! This tutorial focuses on a powerful algebraic technique—s...

TermDefinitionExample System of EquationsA set of two or more equations with the same set of unknown variables. In this context, it often represents two different models or functions describing a single scenario.A company's cost is modeled by C(x) = 10x + 500 and its revenue by R(x) = -x^2 + 80x. The system is { y = 10x + 500, y = -x^2 + 80x }. Substitution MethodAn algebraic method for solving a system of equations by solving one equation for a variable and then substituting that expression into the other equation.Given y = 2x + 1 and 3x + 2y = 16, substitute (2x + 1) for y in the second equation to get 3x + 2(2x + 1) = 16. Point of IntersectionThe point (x, y) where the graphs of two functions cross. This point is the solution to the system of equations representing those functions...

The Substitution Principle If a system is defined by y = f(x) and y = g(x), then the solution occurs where f(x) = g(x). This is the most direct application for systems where both equations are solved for the same variable (often 'y' in function notation). Set the two expressions equal to each other to create a single-variable equation in 'x'. General Substitution Rule For a system of equations, solve one equation for one variable (e.g., solve for y in terms of x). Then, substitute the resulting expression into the other equation. Use this when one equation is easily solved for a variable. This transforms the second equation into an equation with only one variable, which can then be solved.