Substitution Method Calculator

The substitution method is an algebraic technique used to solve a system of two linear equations with two variables. Given a standard system in the form A₁x + B₁y = C₁ and A₂x + B₂y = C₂, the process follows three core steps. First, isolate one variable in the first equation (for example, expressing x in terms of y). Second, substitute this expression into the second equation, replacing that variable to create a single-variable equation. Third, solve for the remaining variable, then back-substitute that result to find the coordinate pair (x, y) that satisfies both equations.

Let us solve the linear system defined by Equation 1: 2x + 3y = 12, and Equation 2: 4x - y = 5. Step 1: Isolate y in the second equation to get y = 4x - 5. Step 2: Substitute this expression into the first equation: 2x + 3(4x - 5) = 12. Step 3: Expand and simplify the equation: 2x + 12x - 15 = 12, which combines to 14x - 15 = 12. Adding 15 to both sides gives 14x = 27, resulting in x = 27/14 or approximately 1.9286. Step 4: Back-substitute x back into our isolated expression for y: y = 4(1.9286) - 5 = 7.7144 - 5 = 2.7144. The final solution is x ≈ 1.9286 and y ≈ 2.7144.

When working with linear systems, always inspect both equations before starting to choose the variable with the smallest coefficient (preferably 1 or -1), as this minimizes fraction usage and arithmetic complexity during isolation. If your equations contain decimals or fractions, clear them early by multiplying the entire equation by the least common multiple of the denominators. Finally, always verify your final coordinate pair by substituting both values back into the original equations to guarantee structural accuracy.