Lesson 2: Substitution

Property

A solution to a $2 \times 2$ linear system is an ordered pair that satisfies both equations.

The substitution method involves solving one equation for a variable, then substituting that expression into the other equation.

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.