AI Tutor
If a smoothie costs two dollars more than a cookie, and two smoothies plus three cookies cost nineteen dollars, you do not need to guess the prices.
A system of linear equations is a set of two or more equations sharing the same variables, and its solution is the exact ordered pair (x,y)(x, y) that satisfies every equation at the same time.
How do we turn this intuitive swap into a foolproof step-by-step algebraic technique?

The Substitution Workflow

The substitution method works by rewriting one equation so that a single variable stands alone, then inserting that expression into the other equation to collapse a two-variable problem into a single-variable equation.
Let us solve the system {y=2x−13x+2y=12\begin{cases} y = 2x - 1 \\ 3x + 2y = 12 \end{cases}.
Since yy is already isolated in the first equation, substitute (2x−1)(2x - 1) into the second equation: 3x+2(2x−1)=123x + 2(2x - 1) = 12.
Distribute the 22 to get 3x+4x−2=123x + 4x - 2 = 12, combine like terms to get 7x−2=127x - 2 = 12, and add 22 to yield 7x=147x = 14, which simplifies to x=2x = 2.
Now back-substitute x=2x = 2 into y=2x−1y = 2x - 1: y=2(2)−1=3y = 2(2) - 1 = 3, giving the final solution (2,3)(2, 3).
What happens when neither variable is already isolated, or when negative signs threaten your distribution?

Strategic Isolation and Distribution