hello in this video we're going to look at the mathematics for Business and Economics dealing with the topic of single equations the basic concept of working with and simplifying equations and the operations performed on an equation must maintain equality the equal sign which means the left-hand side of the equation must still equal the right-hand side of the equation after any mathematical operation so let me give you an example where this doesn't hold so adding a number only to one side of the equation so if we have an equation the left-hand side equals the right-hand side if you were to add a number only to one side of the equation it would no longer be true that the left-hand side equals the right-hand side so you just can't eat add plus 100 to the right-hand side subtracting a number from only one side of the equation will no longer maintain the equality between the left-hand side and the right-hand side so if we were to subtract 50 from the left-hand side it would no longer be true that the left-hand side equals the right-hand side multiplying only one side of the equation will no longer maintain this equality between the left-hand side and the right-hand side for to multiply the right-hand side only by ten doing nothing to the left-hand side the Equality will not be maintained dividing only one side of the equation will also violate maintaining the equality between the left-hand side and the right-hand side here I divide the left-hand side by two it'll no longer be true that the left-hand side equals the right-hand side only squaring or taking the square root of one side of the equation will also be a no-no the left-hand side will no longer equal the right-hand side if we're to only square the right-hand side of the equation so what should we do the key point is what you do to one side of the equation you must do to the other side of the equation in order to maintain the equality so let's look at some examples of maintaining the Equality adding the same number to both sides of the equation like in this case 15 the left-hand side and 15 of the right-hand side will maintain the Equality if we're going to subtract we have to subtract the same number from both sides of the equation so it's okay in this example if we were to subtract say 4 from the left-hand side and 4 from the right-hand side if we were to multiply both sides of the equation by the same number so 2 times the left-hand side will equal 2 times the right-hand side and dividing both sides of the equation by the same numbers also permissible if we're going to square would have to square both sides of the bow sides of the equation the left-hand side squared will equal the right-hand side squared or taking a square root of both sides note here it's okay to multiply one side of the equation by one one times anything is just that anything so 1 times the left-hand side is a is going to maintain the equality here between the left-hand side and the right-hand side alright so what I want to do now is just go through a number of examples of simplifying equations working with just single equations here many of which you will find in economics and business applications so here we have an equation with two variables P and Q we want to solve this for P so we want to isolate the P term so what I'm going to do here is I'm going to add 4p to both sides of the equation so if we add 4p to the both sides on the left hand side we're gonna now have 4p and what about on the right hand side well minus 4p plus 4p will cancel on the right hand side so we now have this the next thing I'll do is I'm going to subtract Q from both sides of the equation and want to get rid of Q here so I'm going to subtract Q from the left-hand side and if I'm subtracting Q from the left-hand side I have to do it to the right-hand side so that'll give us 4 P equals 40 minus Q the next thing I want to do I want to isolate the the P term so we're gonna get rid of this for one way to get rid of this for just divide both sides of the equation through by 4 so 40 divided by 4 and this Q term divided by 4 leaves us with this result and if we want to simplify 40 divided by 4 is just 10 moving on example 2 we got 20 minus 0.5 Q equals 5 let's solve for Q so first thing I'm going to do is subtract 20 from both sides of the equation so on the left hand side we got 20 minus 20 so that's just 0 and on the right hand side we're going to have 5 minus 20 so giving us this result here simplifying the right-hand side 5 minus 20 is minus 15 we got a minus and minus so they're gonna they're gonna cancel or our way we can think about this is we can just simply multiply both sides of the equation by minus 1 so that will leave us with 0.5 Q equals 15 and now let's just isolate 2 Q by dividing both sides by 0.5 so 0.5 divided by 0.5 on the left hand side is 1 and on the right hand side we got 15 divided by 0.5 and that'll just be 30 moving on here's example 3 we have 20 minus 1/4 Q equals 40 plus three-halves Q we we want to solve for Q so first thing I want to do here is I want to subtract 40 from both sides of the equation so on the right hand side 40 - 40 disappears and since we subtracted 40 from the right hand side we must do it to the left-hand side so 200 minus 40 is 160 now we got this minus 1/4 Q let's let's get this to the right hand side so in order to do that we're gonna add 1/4 Q to both sides so on the left hand side minus 1/4 Q plus 1/4 q is just zero leaving us with this on the right-hand side we're going to add the Q terms together on the right hand side we can do this through a common denominator so what I did here is I got the using the common denominator 4 so to turn this denominator 4 I have to multiply it by 2 and since I multiplied the denominator by 2 I have to multiply the numerator by 2 so 3 times 2 is 6 and 2 times 2 is 4 we can add these up now to get 7/4 Q and now let's solve for Q we can multiply both sides of the equation by 4/7 so 7/4 times 4/7 on the right hand side is just 1 leaving this on the left hand side so 160 times 4 is 640 and that's all divided by 7 so we get the answer here of Q equaling 91 point 4 3 moving on example 4 I just want to solve this for X so we've got M and we got there's another this other variable P subscript X we've got X plus 1/3 X we want to solve this for X so the easiest thing here to do is let's factor out ax on the right-hand side of the equation so on the right hand side I'm going to factor out an X term nothing happens to the left hand side so i factor out an X term to give us this result so notice if we're to multiply X by what's in parentheses would just get back the original the P subscript X times X and one-third times X and then the next thing I want to do I want to solve this for X so I want to isolate 2 ax let's divide both sides of the equation through by what's in parentheses so we're going to divide everything through by what's in parentheses doing that we're going to get this result example 5 q equals while this mess we want to solve for Q so the first thing I want to do is simplify up the right-hand side so you're not going to do anything with the 48 first but minus 1/5 times 10 is just minus 10 fifths or minus 2 and then minus 1/5 x minus Q is going to be a positive in this case positive Q divided by 5 r+ q fifths 40 minus 2 is 38 let's we want to isolate 2q so we're going to subtract Q / 5 from both sides so Q divided by 5 minus Q divided by 5 on the right hand side disappears so we're left with 38 and on the left hand side we have this result getting a common denominator here what is Q well Q can be thought of as 5 Q divided by 5 that's just simply Q and then 5 Q divided by 5 minus Q fifths leaves us with 4 Q divided by 5 we want to isolate the Q term again we're going to multiply both sides of the equation by 5/4 why 5/4 because 5/4 times 4/5 on the left hand side will just become 1 and if you do something the one side of the equation you have to do it to the other and we'll get Q equals 38 times 5/4 which is going to be forty seven point five example six we want to solve this for y it's kind of messy here we want to solve this for y so first thing I did here was I'm gonna multiply both sides through by Y divided by four so you'll notice on the right hand side I'm multiplying the right hand side through by Y divided by four and I'm doing that to the left hand side and so what's nice is this y divided by 4 all divided by Y divided by 4 is just going to cancel and become 1 so now we're left with this expression the simplification I did next moving it up over here 8 divided by 4 is 2 so that's just 2y and again I said we want to solve this for y oops so I'm gonna divide both sides through by 2 so dividing the right hand side through by 2 and the left hand side through by 2 we'll get y equals x divided by 4 example seven we want to solve this equation for P so we got this equation and we want to solve it for P so the first thing I do is I add 25 to both sides I just want to isolate the P term so I add 25 to both sides so 25 then it will disappear on the minus 25 will disappear on the right hand side and we'll add 25 to the left hand side next thing I do here is I'm gonna multiply both sides through by 3 so 1/3 times 3 is just 1 so just leaving us with P then 3 times Q is 3 is 3 Q and then 3 times 25 is 75 example 8 we want to solve this equation for K let's solve this for K so I am going to subtract 20 L from both sides that will give us this result just isolating the K term now and now I'm gonna divide both sides through by ten so a thousand divided by ten is a hundred minus 20 divided by 10 is minus two and of course I did that to the left-hand side as well ten divided by ten is one so that's why we just got 1k example nine again let's solve this equation for y there's somewhat of a mess here but we're gonna try to isolate the Y terms so the first thing I did is I multiplied both sides through by a hundred so 100 times the left-hand side will just leave us with this result and a hundred times the right-hand side leaves us with this result all right the next step I'm gonna multiply both sides through by Y raised to the 0.25 power so Y raised to the 0.25 power multiplied by the right hand side the that will cancel just leaving us with a hundred raise 100 times x squared and then on the left hand side we're multiplying that by Y raised to the 0.25 so we get this result following the rules of exponents I'm gonna just add up these exponents here get y equals y raised to the 0.5 power and again I want to isolate the Y term so in order to do that I am going to square both sides so Y raised to the 0.5 squared is just going to be Y and then we got a square the right hand side as well everything on the right hand side so the left hand side simplifies to Y the right hand side 100 squared is 10,000 x squared raised to the second power is just X to the fourth example 10 another equation we want to solve for y so here I kind of did this in two steps I multiplied everything through by a hundred so the hundred cancels on the left hand side but now the 100 appears on the right hand side then I divide it then I multiplied everything through by Y so Y times y and the right hand side will get rid of the y term but Y times y the left hand side is y squared and now again we just want to isolate the Y so we can take the square root of both sides to isolate the Y term so Y squared and the square root of that is just Y the square root of 100 is 10 and x squared taking a square root of x squared is just X okay so rules of exponents we're just going to multiply these 2 times 1/2 and 2 times 1/2 example Loven solving for Q subtracting 12 from both sides the minus signs will cancel or multiplying both sides through by minus 1 gives us this result multiplying both sides through by Q Q's will cancel on the left hand side we got 12 Q and now finally dividing everything through by 12 Q equals 1/2 example 12 solving for Q similar problem to example of and moving 9 over to the right hand side by subtracting 9 from both sides multiplying both sides through by -1 multiplying both sides through by Q squared dividing both sides through by 9 36 divided by 9 is 4 9 divided by 9 is just 1 and then taking the square root of both sides left hand side and right hand side Q squared taking square root of that is just Q and taking a square root of 4 or 4 to the one-half power is just 2 all right here's example 14 we've got one equation and one unknown we want to solve it for Al so the first thing I'm going to do is I'm going to add 16 to both sides minus 16 plus 16 cancels that 16 over here and then 24 plus 16 if now why we got a 40 on the right hand side we want to capture isolate the L terms let's subtract L divided by 12 from both sides so that gives us this result let's get a common denominator so a common denominator here is 12 so 4 times 3 is 12 4 times L is 4 L 4 L divided by 12 minus L divided by 12 is 3 L divided by 12 let's multiply both sides through by 12 divided by 3 so I'm multiplying both sides through by 12 divided by 3 that's just going to give us 160 example 15 solving this for L let's first simplify the left hand side 10 times 8 is 80 10 times negative L or minus L is minus 10 L let's subtract 80 from both sides so 20 minus 80 is why we got minus 60 and getting rid of these minus signs on both sides by multiplying both sides through by minus 1 we get 10 L equals 60 and dividing everything through by 10 L equals 6 and example 16 X divided by 1/3 y equals 4 let's simplify this this mess on the left hand side we can do that by multiplying the left hand side by basically 1 3/3 but if we do that you'll notice that this 3 and the 1/3 will cancel in the denominator just becoming 1 so you get 3 x divided by y equals 4 let's multiply the left hand side and right hand side through by Y okay I'm just showing that here Y divided by Y will cancel become 1 and we'll just get 4y and the right hand side so 3x equals 4y dividing both sides by 3 x equals 4 y divided by 3 say I solve this for X okay I hope you found this video helpful