If a 100 ft tall building is next to a 30 foot tall building and the distance from the top corners of the buildings is 86 ft, how far apart are the buildings? (Hint: think of the distance between the top corners of the buildings as a hypotenuse)

If it helps to visualize this, draw a triangle that starts at the top of the tall building and travels to the top of the smaller building and then back to the tall building creating a right triangle. the side of the triangle touching the tall building can be solved by subtracting the height of the short building form the tall. $$100-30 = 70 $$ since we now know two sides of a right triangle, we can solve for the third using Pythagorean theorem $$a^2 +b^2=c^2$$. $$70^2+x^2=86^2$$ $$4900+x^2=7396$$ $$x^2=2496$$ $$x=\sqrt{2496}$$ $$x=49.95$$ Which Rounds to 50 ft.

What is the difference between Mass and Weight?

Mass is a measure of the how much matter an object has. Weight is actually a force, it is determined by the acceleration on an object using Newtons second law $$ F = m a$$ where F stands for the force (in this case weight), m stands for mass (the amount of matter) and a stands for acceleration (gravity). Mass for an object is constant as long as the object doesn't lose any of its matter, but acceleration can change. This is why you have the same mass whether you are on the earth or the moon but you weigh much lighter on the moon than you do on earth because the acceleration of gravity on the moon is much smaller than it is on earth.

$$5 = \frac{2}{3} x + \frac{1}{3}$$

$$5 = \frac{2}{3} x + \frac{1}{3}$$ First, multiply the 5 by $$\frac{3}{3}$$ to get a common denominator. This is allowed because you are actually multiplying by 1 which doesn't change the left side of the equation but still allows us to work with the fraction better, giving us... $$\frac{15}{3} = \frac{2}{3} x + \frac{1}{3}$$ No we can subtract the $${1}{3}$$ from both sides to get rid of the fraction on the right. (Remember always work on getting the variable by itself first.) This leaves us with.... $$\frac{14}{3}= \frac{2}{3} x $$ Now to get rid of the $$\frac{2}{3}$$ which is being multiplied to the x, we need to divide. But its easier to divide fractions by multiplying by the reciprocal. This just means you take the fraction and switch the top (numerator) and the bottom (denominator) so in this case we multiply both sides of the equation by $$\frac{3}{2}$$ leaving us with... $$\frac{42}{6}=x$$ now we need to simplify, lucky for us $$\frac{42}{6}$$ divides nicely and we get..... $$7=x$$