Solve Systems by Substitution
Solving systems of equations by substitution is a Grade 11 algebra method in Big Ideas Math where one variable is isolated in one equation and substituted into the other, reducing a two-variable system to one variable. For the system y = 2x + 1 and 3x + y = 10: substitute y = 2x + 1 into the second equation: 3x + (2x + 1) = 10 → 5x = 9 → x = 9/5. Then y = 2(9/5) + 1 = 23/5. Substitution is especially efficient when one equation is already solved for a variable. It extends to nonlinear systems where graphical methods give approximate solutions.
Key Concepts
To solve a system of linear equations by substitution: solve one equation for one variable in terms of the other, then substitute this expression into the second equation to create a single variable equation.
Common Questions
What is the substitution method for solving systems of equations?
Solve one equation for one variable, substitute that expression into the other equation, solve the resulting single-variable equation, then back-substitute to find the other variable.
How do you solve y = 2x + 1 and 3x + y = 10 by substitution?
Substitute y = 2x+1 into 3x+y=10: 3x+(2x+1)=10 → 5x=9 → x=9/5. Then y = 2(9/5)+1 = 18/5+5/5 = 23/5. Solution: (9/5, 23/5).
When is substitution more efficient than elimination?
Substitution is efficient when one equation is already solved for a variable (y = ... or x = ...). Elimination is often faster when equations are in standard form with matching coefficients.
How do you check the solution to a system solved by substitution?
Substitute both x and y values back into both original equations and verify both equations are satisfied.
How does substitution work with nonlinear systems (e.g., line and parabola)?
Solve the linear equation for y, substitute into the quadratic, and solve the resulting quadratic equation for x. There may be 0, 1, or 2 solutions.
What does it mean if substitution produces a contradiction like 0 = 5?
A contradiction means the system has no solution—the lines (or curves) are parallel and never intersect.