Physics in schools teaches two contradictory and mutually exclusive things: (1) That the upward lift force on an airplane in flight equal its weight (Lift = Weight = mass x gravity). This is based on applying Newtons 2nd law of motion (F = ma) to the airplane in flight. (2) However, modern commercial airplanes like the Boeing 747–400 can fly with thrust-to-weight ratio as low as 0.3. Here the engine thrust is only 0.3x the weight of the airplane, but this thrust is sufficient to push the airplane forward and generate enough lift to fly. Therefore the upward force required for lift and flight must be a lot less than 0.3x the weight of the Boeing 747-400. (Lift < Thrust < Weight). Both statements cannot be true. Which is correct?
Lift counters Weight. Thrust counters Drag. They are at right angles to each other.
I could explain how airplanes fly, but John S. Denker has the best explanation I know of.
High-school teachers know a lot about how to teach, but occasionally they don't know the material as well as they could.
Added in response to comments:
The video "The physics of flight" was correct up to 2:18. At that point it says "Forward Force + Lift = Thrust". That is not correct. The thrust does not push air downwards, the wings do that simply by moving forward through the air and deflecting it downward. The rest of that video is OK.
Video "The lift paradox" was correct up to 1:27, where it repeats the mistake about thrust being directed upward.
Let me tell you what's really going on. Every airplane has a lift:drag ratio. WW II fighters had a lot of drag, like about 5:1. Modern jets have very little drag, about 25:1. That means, purely gliding with no thrust, a Corsair could glide about 5 miles for every 1 mile descent. A Boeing 737 can glide about 25 miles for each mile descent - it is very efficient. This is really important because the less drag there is, the less thrust and fuel are needed.
Assuming straight and level flight at cruising altitude and speed, if the Corsair weighs/lifts 10,000 lbs, its drag/thrust would be 2,000 lbs or 1/5 of its weight. If the 737 weighs 150,000 lbs, its drag/thrust would be 6,000 lbs or 1/25 of its weight. The more "slippery" a plane is, the lower its drag, and the lower the thrust/fuel needed to cruise.
For planes to be really efficient and use as little fuel as possible, they need to have as high a lift:drag ratio as possible. This brings a price to pay: It's hard to get down! They need special drag-creating speed brakes or spoilers just to be able to lose altitude without gaining too much speed.
Added: I should clarify. The given lift:drag ratio is the best possible for that aircraft, and it's at the best-glide speed, that gives the greatest glide range. That's because drag is broken into two parts, parasitic and induced. Parasitic drag is just from air resistance on the parts that stick out. Induced drag is that due to the wing's angle of attack. The slower the plane is flying, the steeper the AOA has to be, so the greater the induced drag. So when the plane flies slower than the best glide speed the induced drag increases. When it's faster, the parasitic drag increases.
For any airplane to fly level at a constant altitude, the amount of lift generated by its wings must be equal to its weight. If lift exceeds weight, the plane climbs. If weight exceeds lift, the plane descends.
The engine & prop in a light plane like a 4-seat Cessna develops no more than a couple hundred pounds of thrust at takeoff which is much less than the weight of the plane. But that amount of thrust is enough to move the plane's wings through the air at a speed sufficient to make enough lift to get the plane off the ground.
Note that the propulsive thrust does not have to equal the plane's weight in order to make it fly; this is only true if the plane had no wings and you were trying to hoist it off the ground by pointing the plane straight up and "hanging" it on its propeller.
Lift does need to equal weight in order for the plane to fly straight and level. If they are not equal, the plane will be climbing or descending.
Thrust, however, does not have to be equal to lift, even in level flight. Planes are not the only device which has this behavior. Consider a simple inclined plane. If your inclined plane has a a rise of 3:10 (3 feet up for every 10 feet), you'll find that moving a 1 pound mass up that inclined plane only takes 0.3 pounds of force. So there the "thrust" of you pushing the object is quite a lot smaller than the weight of the object.
If you think about it, a plane in straight and level flight is not being moved up or down at all. In an extreme case, a plane sitting on the ground at 0mph has no thrust, and yet is not falling. It's the same game while you're in the air. There's no law which states that any air has to be moving downward to keep the plane afloat. You could have a static body of air underneath the plane (similar to a hovercraft). The only reason airplanes need to push air down is because they're not perfect. Air is always slipping upward past the plane (as gravity pulls the plane downward), so airplanes need to push air downward to make up for that movement.
You seem to be confusing between lift and thrust. They are two different force vectors, pointing in different directions and generated by different causes.
Lift indeed has to be at equal to weight in order for the airplane to maintain level flight. Thrust does not. So, both of the statements you quoted are correct, and there is no contradiction.
The “trick” of airplane flight is that by generating a modest amount of thrust - far less than the weight of the plane, as you noted - we cause the plane to move horizontally, which in turn causes an interaction between the plane’s wings and the air they are moving through that produces another completely different force - lift - that is (during most stages of flight) greater in magnitude than the original thrust force.