Which description best defines the algebraic substitution method for solving a system of equations?

Which description best defines the algebraic substitution method for solving a system of equations?

• A. Solve one equation for one variable and substitute into the other
• B. Use elimination by adding equations
• C. Graph both equations and find intersection
• D. Use matrices to invert coefficients

Answer: (A)

Substitution starts by solving one equation for a single variable, expressing that variable in terms of the others. That expression is then substituted into the other equation, replacing the variable with the expression. This turns the system into a single-variable equation, which you solve, and then you plug back to find the remaining variable(s). For example, with x + y = 3 and 2x - y = 0, solve the first for y: y = 3 - x. Substitute into the second: 2x - (3 - x) = 0 leads to 3x - 3 = 0, so x = 1, and then y = 2. So the solution is (x, y) = (1, 2). This approach is all about isolating a variable and replacing it with an equivalent expression to reduce to a single unknown. Other methods, like elimination (adding equations to cancel a variable), graphing (finding the intersection), or using matrices (a linear-algebra approach), solve the system in different ways.

Substitution starts by solving one equation for a single variable, expressing that variable in terms of the others. That expression is then substituted into the other equation, replacing the variable with the expression. This turns the system into a single-variable equation, which you solve, and then you plug back to find the remaining variable(s).

For example, with x + y = 3 and 2x - y = 0, solve the first for y: y = 3 - x. Substitute into the second: 2x - (3 - x) = 0 leads to 3x - 3 = 0, so x = 1, and then y = 2. So the solution is (x, y) = (1, 2).

This approach is all about isolating a variable and replacing it with an equivalent expression to reduce to a single unknown. Other methods, like elimination (adding equations to cancel a variable), graphing (finding the intersection), or using matrices (a linear-algebra approach), solve the system in different ways.