Lesson 3: Solve Systems by Substitution

Property

To solve a system by substitution, follow these steps:

1. Solve one of the equations for either variable.
2. Substitute the expression from Step 1 into the other equation.
3. Solve the resulting equation.
4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
5. Write the solution as an ordered pair and check that it is a solution to both original equations.

Examples

• Solve the system $y = x + 3$ and $3x + 2y = 19$. Substitute $x+3$ for $y$ in the second equation: $3x + 2(x+3) = 19$. This simplifies to $5x+6=19$, so $5x=13$ and $x=\frac{13}{5}$. Then $y = \frac{13}{5} + 3 = \frac{28}{5}$. The solution is $(\frac{13}{5}, \frac{28}{5})$.
• Solve the system $2x - y = 8$ and $x + 3y = 11$. From the first equation, solve for $y$: $y = 2x - 8$. Substitute this into the second equation: $x + 3(2x-8) = 11$. This gives $7x-24=11$, so $7x=35$ and $x=5$. Then $y=2(5)-8=2$, making the solution $(5, 2)$.

Explanation

This method simplifies a two-variable system into a single-variable equation. By isolating a variable in one equation and plugging its expression into the other, you can solve for one variable and then use that value to find the second.

Property

When using substitution, the variables will sometimes completely cancel each other out. The resulting numerical statement determines the number of solutions and the geometric classification of the system:

• Identity (True Statement): If solving results in a true statement like $0 = 0$ or $5 = 5$, the system has Infinitely Many Solutions. The equations are dependent (they represent the exact same coincident line).
• Contradiction (False Statement): If solving results in a false statement like $0 = -10$ or $2 = -1$, the system has No Solution. The equations are inconsistent (they represent parallel lines).

Examples

• Infinite Solutions (Identity): Solve $y = 2x - 3$ and $4x - 2y = 6$.

Substitute $y$: $4x - 2(2x - 3) = 6$.
Distribute: $4x - 4x + 6 = 6$.
Simplify: $6 = 6$. This is a true statement, so there are infinitely many solutions.

• No Solution (Contradiction): Solve $y = 5x + 2$ and $y = 5x - 1$.

Substitute $y$: $5x + 2 = 5x - 1$.
Subtract $5x$ from both sides: $2 = -1$. This is a false statement, so there is no solution.

Explanation

If your variables vanish during substitution, the algebra is trying to tell you something about the geometry of the lines! A true statement like $6 = 6$ means the two equations are actually identical twins disguised as different math problems; they overlap perfectly. A false statement like $2 = -1$ means you have reached a mathematical dead-end because the lines are parallel and will literally never cross.