Back-Substitution in Systems
Back-substitution is the final step of the substitution method for solving systems of equations, as taught in Grade 11 enVision Algebra 1 (Chapter 4: Systems of Linear Equations and Inequalities). After solving for one variable, substitute that value into either original equation to find the second variable. If x was solved first, substitute into any equation containing y to get y. Always use the simpler equation to minimize errors. This completes the ordered pair (x, y) solution.
Key Concepts
Back substitution is the process of finding the first variable's value after solving for the second variable. If you solved for $x$ first, substitute that value into either original equation to find $y$. If you solved for $y$ first, substitute that value to find $x$.
Common Questions
What is back-substitution in solving systems of equations?
Back-substitution is plugging the already-found value of one variable into an original equation to solve for the remaining variable.
Which equation should you use for back-substitution?
Use either original equation — choose the simpler one with smaller coefficients to reduce arithmetic errors.
When does back-substitution occur in the substitution method?
It is the last step: after you solve for one variable, substitute its value back to find the other, completing the ordered pair solution.
What is the result of back-substitution?
A specific numerical value for the second variable, giving you the complete solution as an ordered pair (x, y).
What is a common error in back-substitution?
Substituting into the already-simplified equation rather than an original equation, or substituting into the equation used to solve the first variable (this can lead to circular results).
Can you substitute back into any of the original equations?
Yes. Both original equations share the same solution, so substituting the known variable value into either one will give the correct second value.