Skill: Choose an efficient method for solving a linear system

Before solving, look at the structure of the equations:

• Substitution is efficient when one variable is already isolated.
• Elimination is efficient when the coefficients of one variable match or can be made to match easily.
• Graphing is useful when you want to see the intersection or estimate a solution.

Example

Solve the system:

$$\begin{cases} y=2x+1\\ 3x+y=11 \end{cases}$$

1. Choose a method

The first equation already has $y$ isolated, so substitution is the most efficient method.

2. Substitute

The expression $2x+1$ equals $y$. Replace $y$ in the second equation with $2x+1$:

$$3x+(2x+1)=11$$

3. Solve for $x$

Combine like terms:

$$5x+1=11$$

Subtract $1$ from both sides:

$$5x=10$$

Divide by $5$:

$$x=2$$

4. Find $y$

Substitute $x=2$ into the first equation:

$$y=2(2)+1=5$$

So the solution is:

$$\boxed{(2,5)}$$

5. Verify the solution

Check $(2,5)$ in both original equations:

$$5=2(2)+1$$

This is true. Also,

$$3(2)+5=11$$

This is true.

Therefore, the lines intersect at $(2,5)$, so the system has one solution. Equivalent algebraic steps do not change the common solution.