Skill: Solve systems of equations by substitution

Substitution works by replacing one variable with an equivalent expression.

Consider the system:

$$\begin{aligned} 2x+y&=11\\ x-y&=1 \end{aligned}$$

1. Isolate one variable

Use the first equation and solve for $y$:

$$2x+y=11$$

Subtract $2x$ from both sides:

$$y=11-2x$$

Now we know that $y$ is equivalent to $11-2x$.

2. Substitute into the other equation

The second equation is:

$$x-y=1$$

Replace $y$ with $11-2x$:

$$x-(11-2x)=1$$

3. Solve for the remaining variable

Distribute the negative sign:

$$x-11+2x=1$$

Combine like terms:

$$3x-11=1$$

Add $11$ to both sides:

$$3x=12$$

Divide by $3$:

$$x=4$$

4. Find the other variable

Substitute $x=4$ into $y=11-2x$:

$$y=11-2(4)$$ $$y=11-8=3$$

So the solution is:

$$\boxed{(4,3)}$$

5. Check the solution

Substitute $x=4$ and $y=3$ into both original equations:

$$2(4)+3=11$$ $$4-3=1$$

Both equations are true, so $(4,3)$ is the solution. It represents the point where the two lines intersect.