Section 5.2: Solving Systems of Linear Equations by 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.

Property

When solving systems by substitution, choose to solve for a variable that has a coefficient of $1$ or $-1$ to minimize algebraic manipulation and reduce errors.

Examples