Can I lose weight by eating 1,300 calories a day?
Yes, but you'll have a harder time than eating 1600cals. (Assuming you are a 70kg, sedentary person; if you are a 150cm 50kg person, you are still in the good range with 1300, even if it's for a hair).
What? How can that be?
Because with 1300–1200, you'll put your body eventually in starvation mode. And in starvation mode, well, your body consume much less energy. Is a save mode. It makes you less energetic, moodier, lowers your defenses and crush your libido. All that are system shutting down.
The secret to losing fat is eating 500cals less than your maintenance calories. That is around 1600 cals for a standard person (70Kg sedentary).
That puts you in a deficit good enough to lose fat, but no strong enough to raise metabolic elasticity (starvation mode).
Go to low in the caloric intake, and your body is going agaisnt you, because you are lowering your metabolism... and your losing of weight will be always relative to your metabolism. The greater ΔVo is, most weight will you lose, that is... the greater your total volume of Oxygen breathed in a given time, the greater weight you'll lose.
Because you lose weight brething. Each 02 molecule you inhale will be exhaled as one molecule of CO2. That carbon atom is the weight you lose.
80% of weight loss is through the breath. Surprise!
That's why you also need exercise: to enhance your breathing rate (not only for the duration, but your metabolic base rate) and lose weight faster.
And eating 1300 cals (except if you are a 50kg-150cm kind of guy) will slow your metabolism, your breath, and as such, by basic conservation of mass, your weight loss.
Weight loss is a compromise between raise your metabolic level and lower your calorie input. Going to extremes in caloric intake never, ever, works.