1.3.2 Substitution Method

Solve systems of linear equations by isolating one variable and substituting it into another equation.

定义

The Substitution Method is an algebraic technique for solving systems of linear equations. It involves three main steps: (1) Solve one equation for one variable in terms of the other variable(s), (2) Substitute this expression into the other equation(s) to eliminate that variable, and (3) Solve the resulting equation for the remaining variable(s), then back-substitute to find all variable values. For a system of two linear equations with two unknowns, such as \(\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\), the substitution method reduces the system to a single equation in one variable, making it solvable through standard algebraic techniques.

核心公式

• \(y = mx + b\) (Slope-intercept form for isolating a variable)
• \(\text{If } y = f(x), \text{ then substitute } y \text{ into the second equation: } a_2x + b_2f(x) = c_2\)
• \(\text{General substitution: If } x = g(y), \text{ then } a_1g(y) + b_1y = c_1\)
• \(\text{Solution verification: Substitute }(x_0, y_0)\text{ into both original equations to confirm } a_1x_0 + b_1y_0 = c_1 \text{ and } a_2x_0 + b_2y_0 = c_2\)
• \(\text{For dependent systems: If after substitution we get } 0 = 0, \text{ the system has infinitely many solutions; if we get } 0 = k \text{ (where } k \neq 0), \text{ the system has no solution}\)

易错点

• ⚠️ Algebraic errors when isolating the variable: Students often make sign errors or forget to apply operations to both sides of the equation when solving for one variable. For example, isolating \(y\) from \(2x + 3y = 6\) as \(y = 2x + 6\) instead of \(y = rac{6-2x}{3}\).
• ⚠️ Failing to back-substitute: After finding the value of one variable, students forget to substitute it back into one of the original equations to find the other variable, leaving the solution incomplete.
• ⚠️ Incorrect substitution into the second equation: When substituting the expression for one variable into the second equation, students may make errors in distributing coefficients or combining like terms. For example, substituting \(y = 2x + 1\) into \(3x + 2y = 10\) as \(3x + 2(2x + 1) = 10\) but then incorrectly simplifying to \(3x + 4x + 1 = 10\) instead of \(3x + 4x + 2 = 10\).
• ⚠️ Not verifying the solution: Students fail to check their final answer by substituting both values back into both original equations, missing errors that occurred during the solving process.