Solving Systems by Substitution
Solve systems of two linear equations using the substitution method — isolate one variable, substitute into the second equation, and find both solutions in Grade 9 Algebra.
Key Concepts
Property The substitution method involves solving one of the equations for one variable and then substituting the result into the second equation to solve for the second variable.
Given a system of two equations in two variables, solve using the substitution method. 1. Solve one of the two equations for one of the variables in terms of the other. 2. Substitute the expression for this variable into the second equation, then solve for the remaining variable. 3. Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair. 4. Check the solution in both equations.
Examples For the system $y = 2x + 1$ and $3x + y = 11$, substitute $2x + 1$ for $y$ in the second equation: $3x + (2x + 1) = 11$, which gives $5x = 10$, so $x = 2$. Then $y = 2(2) + 1 = 5$. The solution is $(2, 5)$. Solve the system $x y = 4$ and $2x + 3y = 18$. From the first equation, $x = y + 4$. Substitute into the second: $2(y + 4) + 3y = 18$, so $5y + 8 = 18$, giving $y = 2$. Then $x = 2 + 4 = 6$. The solution is $(6, 2)$. In the system $2a + b = 5$ and $a 2b = 5$, solve the first for $b$: $b = 5 2a$. Substitute: $a 2(5 2a) = 5$, so $5a 10 = 5$, which means $a = 3$. Then $b = 5 2(3) = 1$. The solution is $(3, 1)$.
Common Questions
What is the substitution method for solving systems of equations?
The substitution method involves solving one equation for one variable, then substituting that expression into the second equation. This reduces the system to one equation with one variable, which you solve and substitute back to find both values.
When should you use substitution instead of graphing or elimination?
Substitution works best when one equation is already solved for a variable (like y = 2x + 3) or when a variable has a coefficient of 1, making it easy to isolate. It gives exact answers unlike graphing, which relies on estimating intersection points.
How do you check your solution to a system solved by substitution?
Substitute both x and y values back into both original equations. If both equations are true statements, your solution is correct. An ordered pair (x, y) is the solution only if it satisfies every equation in the system.