How do solve this equation with dot product?

Question

$$[(2,-7)+t(2,10)]\cdot n_1=0$$

$$\text{Solving for $t$, we find }t=\dfrac5{12}$$

$n_1=(1,1)$ or $(-1,-1)$.

How does $t=\dfrac5{12}$?

Answer 1

$$(2+2t,-7+10t)\cdot (1,1)=2+2t-7+10t=0.$$ Then $-5=-12t$ and the result follows. Same for $(-1,-1)$.

Answer 2

This line tells you with $n_1=(1,1)$ or $n_1=-(1,1)$ that $$2+2t=-(-7+10t)$$ Which is simply $$12t=5$$ and thus $$t=\frac{5}{12}$$