Find the slope and y-intercept of : -2y-10+2x=0
1) So the first step is to realize that this is not in slop-intercept form (which is y=mx+b where "m" is the slope and "b" is the slope intercept). We will try to manipulate this equation to turn it into slope-intercept form 2) As you can see, slope-intercept form has the variable "y" by itself on one side of the equation line, so we will replicate this with our equation. 3) To get "y" by itself, we will isolate the variable by removing the "-10" and "2x" from the side it is on to the other. 4) To do this, we will add 10 to both sides in order to remove it from the left side, and subtract 2x from both sides to remove it from the left side. 5) The resulting equation is -2y=-2x+10. 6) Now, given that the "y" still has a coefficient that is not one, we know that we are not yet done. The next step is to multiply both sides by the inverse of -2, which is -1/2 (an inverse is the number that, when multiplied to the original, the product is 1. For example, if the number was 5, the reciprocal would be 1/5. If the number was ⅓, the reciprocal would be 3. If the number was 2/5, the reciprocal would be 5/2.) 7) After multiplying both sides by -½, the resulting equation is y= -1/2(-2x+10). 8) Now all we do is distribute the -½ and multiply it to the -2 and the 10. 9) the resulting equation is y= x-5. 10) Comparing this to the y=mx+b form, we see that the slope is the coefficient of x (which in this case is 1) and the y-intercept is -5. So Slope=1 and y-intercept= -5
Solve for z: 2z + 8 = 2 + 5z
1) In this equation, the first step we want to take is to get like terms, or terms with the same variables, on the same side of the equation sign. Here we have the variable "z" and we have constants, therefore, we will get the z's (2z and 5z) on one side, and the constants (8 and 2) on the other. 2) To do this, we can move the Z's and the constants to either side, as long as they are on different sides. This means that the Z's can be on the left side of the equation side, or the right, as long as the constants are on the other side. I will move the Z's to the right side, and the constants to the left. 3)To do this, we see that the 2z and the 2 are on the wrong side, therefore we will move them to the other. First, let's subtract the 2z from the left side, and, in order to keep the equation balanced, we will also subtract it from the right side. This will eliminate the 2z from the left side and put a "-2z" on the right 4) the resulting equation is 8=2+5z-2z. 5) Then, we will do essentially the same thing to the 2 that is on the right side. We will subtract it from the right side (and of course that means we subtract it from the left in order to keep the equation balanced), thus removing the 2 from the left side. 6) The resulting equation is 8-2=5z-2z. 7) Now, we will simplify by combining like terms, which, in this case, means we subtract 2 from 8 and 2z from 5z. Now, when subtracting quantities that have variables (such as 5z and 2z), all we do is subtract their coefficients. This means that 5z-2z would equal 3z. -Extra examples of this are: •16x-12x=4x • 8m+12m=20m •20xy-14xy=6xy 8) After combining like terms, the resulting equation is 6=3z. 9) Now, in order to solve for z, we need to make sure that its coefficient is 1, so we will multiply it by the inverse of its coefficient of 3, which is ⅓. In order to keep the equation balanced, we will also multiply the 6 by ⅓. -the coefficient is simply the number that, when multiplied to another, will have a product of 1 Examples are: • inverse of 2 is 1/2 • inverse of -1/5 is 5 • inverse of 3/2 is ⅔ 10) After multiplying 6 by ⅓, the resulting equation is z=2 which gives us our answer.
Why is demand downward sloping?
It is because the law of demand dictates that, all things equal, there is an inverse relationship between price and quantity demanded. Therefore, as price increases, the quantity demanded decreases, and as price decreases, the quantity demanded increases. We can see this in everyday life. If you see that there is a sale at a store that you like, you will tend to buy more clothes than you would if the clothes were at full price. In fact, you might have had no intention of shopping at all, but upon hearing that there is a sale, you were willing to go and buy. This phenomenon of consuming more as the price decreases is outlined by the law of demand, and explains why demand is downward sloping.