Skill: Solve systems of linear equations algebraically

To solve a system of linear equations algebraically, find the values of $x$ and $y$ that make both equations true. One useful method is elimination, which combines the equations to eliminate one variable.

Consider the system

$$\begin{cases} 2x+3y=13\\ 4x-y=5 \end{cases}$$

Step 1: Eliminate one variable

The second equation has $-y$, while the first has $+3y$. Multiply the second equation by $3$:

$$3(4x-y)=3(5)$$

This gives

$$12x-3y=15$$

The system is now

$$\begin{cases} 2x+3y=13\\ 12x-3y=15 \end{cases}$$

Step 2: Add the equations

Adding the equations eliminates $y$:

$$\begin{aligned} 2x+3y&=13\\ 12x-3y&=15\\ \hline 14x&=28 \end{aligned}$$

Solve for $x$:

$$x=2$$

Step 3: Substitute to find the other variable

Substitute $x=2$ into either original equation. Using $4x-y=5$:

$$4(2)-y=5$$ $$8-y=5$$ $$-y=-3$$ $$y=3$$

Therefore, the solution is

$$\boxed{(2,3)}$$

Step 4: Verify in both original equations

Check $(2,3)$ in the first equation:

$$2(2)+3(3)=4+9=13$$

Check it in the second equation:

$$4(2)-3=8-3=5$$

Both equations are true, so $\boxed{(2,3)}$ is the solution. This system has a unique solution because the variables led to one specific ordered pair.