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We're still on chapter 2.
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Section 2.4 part 1, solve using algebra to find a variable, write an equation, solve,
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and check.
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So the sum of the measures of the angles in a triangle is 180 degrees.
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Two angles have the same measure and the third angle is 20 degrees less than the sum
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of the two smaller angles.
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Find the measure of the three angles.
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So we're looking for three angles.
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Two of them are actually the same.
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So let's just say the first angle, the second angle, and then the third angle.
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It says two of them have the same measure.
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So let's let those be the first two.
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Let's call them x and x. Later we'll put in the degree symbols.
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And the third one is 20 degrees less than the sum of the two smaller.
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So that means I have to add the two smaller and then subtract 20.
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So the third one is really 2x minus 20.
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So we have three angles and what do we know about the sum of the measures of the angles
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in a triangle that they add up to 180 degrees.
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So we're going to add the three angles together.
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Here's the first angle plus the second angle plus the third angle is 180 degrees.
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Notice I'm leaving off the degree symbol because it's easier to solve the equation without
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using the degrees, but later we will put that in for our answer.
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All right, so here's how we define our variables and we have an equation.
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So now we solve it.
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So we have x plus x plus 2x.
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So we've got to add those like terms.
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That's 4x minus 20.
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And then we'll have to add 20 to both sides to isolate the variable on the left side
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of the equation for x equals 200.
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And then divide by 4 to finally get x equals 50.
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So it looks like x is 50.
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So it looks like the first angle would be 50.
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So it's right 50 degrees and this would be the same one.
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It would be 50 degrees.
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Let's see, if x was 50 degrees, the first two are 50.
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It says the third one is 20 less than twice that or add the two and minus 20.
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So that would be 100 minus 20 or 80 degrees.
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And does 50, 50 and 80 add up to 180 degrees?
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Yes, it does.
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Awesome.
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So the answer is the angles measure 50 degrees, 50 degrees, and 80 degrees.
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And the check, of course, if you need to write that out, you could just write 50, 50,
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and 80.
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That's up to 180.
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And you're also making sure that this third angle really is 20 less than the sum of the
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first two.
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50 plus 50 is 100 minus 20 really does add up to 80 degrees.
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So it all makes sense.
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Removing on to section 2.5, part 1, substitute the given values into each given formula
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and solve for the unknown.
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So the first formula is a equals 1 half h in parentheses b plus b.
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This is actually the formula for the area of a trapezoid.
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What's important to note here is there is an upper case b in a lower case b and those
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represent different numbers.
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So when you are doing math, be very careful not to switch between lower case and upper
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case letters.
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They may represent different things.
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All right, so what we need to do is solve for the variable that isn't here and that's
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h because it didn't tell me what h is.
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I know what a is.
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I know what the upper case b, the lower case b is.
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So we are going to plug all the numbers in.
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Instead of a, we are going to write 52, replacing a with 52.
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We don't know h, so we will just write h, 1 half h and the large b is 8, the little b
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is 5.
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So the first step is to replace the variables with the given values and then we need to
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solve for h.
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All right, at this point I would simplify the right hand side first by adding 8 and 5
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inside the parentheses.
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So we have 52 equals 1 half h times 13.
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And it does look funny to see that variable in the middle.
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So really this is 1 half times 13 is the coefficient.
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So you have 52 equals, you can just write that as 13 halves h, a little bit easier to deal
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with writing it that way.
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Now how do you solve for h when you have 13 halves times h?
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One way is multiplied by the reciprocal.
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So the reciprocal of 13 halves is 2 13.
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So if I multiply both sides by 2 13's, then the 13's cancel and the t's cancel over
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here.
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And low and behold, 13 goes into 52 four times.
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I remember that because I think of a deck of cards.
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You've got 13 in each suit and there's 52 cards in a deck.
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So I know 13 times 4 is 52.
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So 2 times 4 is h.
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So 8 equals h.
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And that's exactly what we're looking for.
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So the answer is h equals 8.
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Now to check this, what you could do is plug in 8 for h, 8 for the large b, 5 for the
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small b, and simplify the right hand side and see if you would get 52 for the area.
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The next formula is p equals 2l plus 2w.
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This is the formula for the perimeter of a rectangle.
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And what we're given this time is p and l.
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We're given the perimeter and length.
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So what we don't know is the width.
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W would be standing for the width.
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So the first step is to replace the variables with the given values.
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So in place of p will write 96.
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And in place of l will write 26.
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And we've got plus 2w.
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And so now we're going to solve this equation.
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We only have one unknown, the w.
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So we need to first do the multiplication on the right hand side since it isn't simplified.
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So 2 times 26 is 52 plus 2w.
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Now you want to isolate tw.
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So you could subtract 52 from both sides.
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That gives you 44 equals tw.
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And if I buy 2.
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So we get 22 is w.
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I'm going to turn that around and write it as w equals 22.
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It looks like that's going to be the answer.
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Now for this one, let's look at a rectangle and see if that would make sense.
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See if you believe this.
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If the length was 26 and we just decided that the width was 22, then this is what the
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rectangle would look like.
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And convince yourself that if you add the four sides you would get 96 or go ahead and
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add it.
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22 and 26 and 22 and 26.
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Just add the four sides.
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That's how you get the perimeter of anything.
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You add the lengths of the sides.
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So let's see, 8 and 8 is 16 and 96.
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So actually it checks as well.
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You can look at the picture.
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We're going on to 2.5.
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Part 2.
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Solve each formula for the specified variable.
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So number 1, c equals 2 pi r.
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We're going to solve for r.
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By the way, this is the formula for the circumference of a circle.
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We're c's the circumference and r is the radius.
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So if this was just c equals 2 r, you would just divide by 2, the coefficient.
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But instead of 2, it's slightly more complicated.
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The coefficient is 2 pi.
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But we do exactly the same thing.
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You need to isolate r so you just divide by the coefficient of 2 pi.
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Sorry, 2 pi.
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So c over 2 pi is r and that's the answer.
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I've just solved for r c over 2 pi.
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What's that?