Solve the system y = x + 5 and y = 2x - 7

Expert intro: Both equations are already written as expressions for $y$, so set them equal to each other. That gives a direct path to the intersection point.

Detailed walkthrough

We solve

$y=x+5$

$y=2x-7.$

Since both equal $y$, set the right-hand sides equal:

$x+5=2x-7.$

Subtract $x$ from both sides:

$5=x-7.$

Add 7 to both sides:

$x=12.$

Now substitute into either equation:

$y=12+5=17.$

So the solution is

$\boxed{(12,17)}.$

💡 Pitfall guide

A common error is mixing up the variables and solving for $y$ first when both equations already give $y$ directly. Setting the expressions equal is the cleanest method here.

🔄 Real-world variant

If one equation were instead in terms of $x$, you could still solve by substitution or elimination. The key idea is that the intersection point must satisfy both equations at the same time.

🔍 Related terms

intersection point, substitution, linear equations