WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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Today is our first lesson for the Algebra II series, and we are going to start out with some review of concepts from Algebra I.
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If you need more detail about any of these concepts, please check out the Algebra I series here at Educator.
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The first session is on expressions and formulas.
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Recall the earlier concepts of variables and algebraic expressions:
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starting out with some definitions, a **variable** is a letter or symbol that is used to represent an unknown number.
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It could be any letter; frequently, x, y, and z are used, but you could choose n or s or w.
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**Algebraic expressions** means that terms using both variables and numbers are combined using arithmetic operations.
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Remember that a **term** is a number, or a variable, or both.
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So, a term could be 4--that is a constant, and it is a term; it could be 2x; it could be y².
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And when these are combined using arithmetic operations, then they are known as expressions.
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And when variables are involved, then they are algebraic expressions.
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For example, 4x³+2xy-1 would be an example of an algebraic expression.
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The rules specifying order of operations are very important; and they are used in order to evaluate algebraic expressions.
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Recall the procedure to evaluate an expression using the order of operations.
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First, evaluate expressions that are inside grouping symbols: examples would be parentheses, braces, and brackets.
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The next thing, when you are evaluating an algebraic expression, is to evaluate powers.
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So, if a term is raised to a power (such as 4² or 3⁴), you need to evaluate that next; that is the second step.
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Next is to multiply and divide, going from left to right.
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You start out at the left side of an expression; and if you hit something that needs to be divided, you do that.
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And you proceed towards the right; if you see something that needs to be multiplied, you do that.
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It is not "multiply all the way, and then go back and divide"; it is "start at the left; any multiplication or division--do it."
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Move to the next step; move towards the right; multiply or divide...and so on, until all of that has been taken care of.
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Finally, you do the same thing with addition and subtraction: you add and subtract from left to right.
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And we will be illustrating these concepts in the examples.
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One thing to recall is that a fraction bar can function as a grouping symbol.
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For example, if I have something like 3x-2x+2, all over 4(x+3)+3, I would treat this as a grouping symbol.
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And I would simplify this as far as I could, going through my four steps; and then I would simplify this;
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and *then* I would divide this simplified expression on the top by the simplified expression on the bottom.
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And remember, the reason that we use order of operations is that, if we didn't, and everybody was just doing things their own way,
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we couldn't really communicate using math, because people would write something down,
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and somebody might do it in a different order and come up with a different answer.
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So, this way, it is an agreed-upon set of rules that everyone follows.
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Monomials: a **monomial** is a product of a number and 0 or more variables.
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Again, refreshing your memory from Algebra I: examples of a monomial would be 5y, 6xy², z, 5.
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So, it says it is a product of a number and zero or more variables.
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Here, there aren't any variables, so that actually is simply a constant; but it is still called a monomial, also.
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Here, I have 5 times one variable; here I have multiple variables.
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This is examples of...these are all monomials.
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A **constant** is simply a number; so, it could be -3 or 6 or 14; those are constants.
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Coefficients: a **coefficient** is the number in front of the variable.
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Up here, I said I had 5y and 6xy²: this is a coefficient: 5 is a coefficient, and 6 is a coefficient.
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When you see something like z, it does have a coefficient: it actually has a coefficient of 1.
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However, by convention, we usually don't write the 1--we just write it as z, but it actually does have a coefficient of 1.
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Next, degree--the degree of a monomial: the **degree** of a monomial is the sum of the degrees of all of the variables.
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So, it is the sum of the degree of all of the variables.
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For example, 3xy²z⁴: if I want to find the degree, I am going to add the degree of each variable.
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This is x (but that really means x to the 1--the 1 is unstated) plus y² (the degree is 2), plus z⁴ (the degree is 4).
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Adding these up, the degree for this monomial is 7.
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When we talk about powers: **powers** refer to a number or variable being multiplied by itself n times, where n is the power.
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For example, if I say that I have 5², what I am really saying is 5 times 5.
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So, 5 is being multiplied by itself twice, where n equals 2.
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I could say I have y⁴: that equals y times y times y times y; and here, the power is 4.
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OK, continuing on with more concepts: a **polynomial** is a monomial or a sum of monomials.
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Recall the concepts of term, like terms, binomial, and trinomial.
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A polynomial is simply an expression in which the terms are monomials.
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And we say "sum," but this applies to subtraction, as well--a polynomial can certainly involve subtraction.
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For example, 4x²+x or 2y²+3y+4: these are both polynomials.
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We also could say that 5z is a polynomial, but it is also a monomial; there is only one term, so it is a polynomial, but we usually just say it is a monomial.
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OK, so looking at these other words: a **binomial** is a polynomial that contains two terms.
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So, it is the sum of two monomials, whereas the **trinomial** is the sum of three monomials.
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A monomial is simply a single monomial.
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Recall that, as discussed, a **term** is a number or a letter (which is a variable) or both, separated by a sign.
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Terms could be a number; they could be a variable; or they could be both.
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3x-7+z: here, I have a number and a variable; here, I just have a number (I have a constant); here, I just have a variable.
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And they are separated by signs--by a negative sign and a positive sign--so each one of these is a term.
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The concept of **like terms** is very important, because like terms can be added or subtracted.
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Like terms contain the same variables to the same powers.
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For example, 1 and 6 are like terms; they don't contain any variables, so they are like terms.
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3xy and 4xy are like terms; they both contain an x to the first power and a y to the first power.
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2y² and 8y² are also like terms: they both contain a y raised to the second power.
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And so, these can be combined: they can be added and subtracted.
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A **formula** is an equation involving several variables (2 or more), and it describes a relationship among the quantities represented by the variables.
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And we have worked with formulas previously: and just to review, one formula that we talked about is the Pythagorean Theorem.
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That is a²+b²=c², where c is the length
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of the hypotenuse of a right triangle, and a and b are the lengths of the two sides.
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And this tells us the relationship among the three sides of the triangle.
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And that is really what formulas are all about, and really what algebra is all about: describing relationships between various things.
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And of course, during this course, we are going to be working with various formulas.
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OK, in this example, we are asked to simplify or evaluate an algebraic expression.
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5x²...and they are telling me x=3, y=-3; so I have some x terms and some y terms.
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My first step is to substitute: so, everywhere I have an x, I am putting in a 3; everywhere I have a y, I am putting in a -3.
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So, here I have 3xy, so here it is going to be 3 times 3 times -3.
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Recall the order of operations: the first thing I am going to do is to get rid of the grouping symbols.
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Take care of the parentheses; and looking, I do have parentheses.
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In here, I have a negative and a negative; so I am simplifying that just to positive 3.
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OK, continuing to simplify inside the parentheses: 3 plus 3 is 6.
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I completed my first step in the order of operations.
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The next thing to do is evaluate powers; and I do have some terms that are raised to various powers.
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3² is 9, minus 2 times 6³; so, 6 times 6 is 36, times 6 is 216.
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That took care of my powers; and the next thing is going to be to multiply and divide.
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And when we do that, we always proceed from left to right.
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2 times 216 is 432; OK, now I have: 3 times 3 is 9; 9 times -3 is -27.
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I am going to rewrite this as 9 minus 432 minus 27.
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Finally, add and subtract; and this is going from left to right.
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9 minus 432 gives me -423, minus 27 (so now I have another bit of subtraction to do--that is -423-27) gives me -450.
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So again, the first step was substituting in 3 and -3 for x and y.
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The next step was to get rid of my grouping symbols; evaluate the powers;
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multiply and divide, going from left to right (and I just had multiplication here);
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and then, add and subtract, going from left to right, to get -450.
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In this second example, again, we are asked to evaluate an algebraic expression; and here, we have three variables: a, b, and c.
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So, carefully substituting in each of these, a=-1...so -1², minus 2, times b (b is 2), times c (c is 3), plus 3³.
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Here, I have c² in the denominator; so that gives me 3², minus 2, times a (which is -1), times b (which is 2).
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Since there was a lot of substituting, it is a good idea to check your work.
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a is -1 (that is -1²), minus 2, times b, times c, plus c³;
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all of that is divided by 3² (so that is c²) minus 2, times a, times b.
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Everything looks good; now, the first thing I want to do is eliminate grouping symbols.
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Recall that, in this type of a case, the fraction bar is functioning as a grouping symbol.
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So, the whole numerator should be simplified, and the denominator should be simplified; and then I should divide one by the other.
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Starting with the numerator: within the numerator, there are not any grouping symbols,
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so I am going to go ahead and go to the next step, which is to take care of powers.
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And -1 times -1 is 1; and then, I have 3³; that is 3 times 3 (is 9), times 3 (is 27).
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OK, I can do the same thing in the denominator; I can just do these both in parallel.
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And so, I am going to evaluate the powers in the denominator.
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3 times 3 is 9; and then, I don't have any more powers--OK.
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So, I took care of that; my next step is going to be to multiply and divide.
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OK, so I have, in the numerator, 1 minus 2 times 2 (is 4), and then 4 times 3 (is 12), plus 27.
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So, that took care of that step; now, in the denominator, I have 9, and then I have minus 2, times -1.
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So, 2 times -1 is going to give me -2; -2 times 2 is going to give me -4.
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OK, so now, I have taken care of all of the multiplication and the division.
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The next step is to add and subtract--once again, going from left to right.
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So, starting up here, the next step is going to be 1 minus 12; 1 minus 12 is going to give me -11.
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So, it is -11 plus 27; that is going to leave me with 16 in the numerator.
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In the denominator, I have 9, minus -4; well, a negative and a negative gives me a positive,
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so in the denominator, I actually have 9 plus 4, which gives me 13.
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The result is 16 over 13.
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Again, starting out by substituting values for a, b, and c...I have done that in this first step.
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And then, I treat this fraction bar as a big grouping symbol, and then I take care of the numerator and the denominator separately.
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You could have done them one at a time, or you can do steps at the same time.
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So, first, evaluate the powers; I did that in the numerator; I did that in the denominator (I am treating them separately).
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Multiplying and dividing: I did my multiplication here and in the denominator.
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And finally, adding and subtracting to get 16/13.
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Example 3: The formula for the area of a triangle is Area equals 1/2 bh.
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So, this is actually that the area equals one-half the base times the height of the triangle.
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Find the height if a is 32, and the base (b) is 8.
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OK, here we are being asked to find the height, and we are given the other two variables.
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So, let me rewrite the formula: area equals 1/2 base times height.
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Now, I am going to substitute in what I was given.
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I am given the area; I am given the base; and I need to find the height.
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What I need to do is isolate h; so, first simplifying this: 32=...well, 1/2 of 8 is 4, so that gives me...4h.
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Next, divide both sides of the equation by 4 (32/4 and 4h/4) in order to isolate that.
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Well, 32 divided by 4 is 8; the 4's cancel out on the right; and then just rewriting this in a more standard form, with the variable on left, the height is 8.
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So again, first just write out the formula; substitute in a and b (which I was given).
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And then, solve for the height.
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The temperature in Fahrenheit is F=9/5C+32, where C is the temperature in Celsius.
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If the temperature is 78 degrees Fahrenheit, what is it in Celsius?
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Rewrite the formula and substitute in what is given.
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What is given is that the temperature in Fahrenheit is 78.
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And I am looking for Celsius (I always keep in mind what I am looking for--what is my goal?).
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And that is +32; my goal here is going to be to solve for C--to isolate that.
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Subtracting 32 from both sides gives me 46=9/5C.
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Now, in order to isolate the Celsius, I am going to multiply both sides by 5/9.
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When I do that, I am going to get 5 times 46 (is 230), and that is divided by 9.
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Here, that all cancels out; so, rewriting this, Celsius equals 230/9.
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That is not usually how we talk about temperature; so simplifying that, if I took 230 and divided it by 9, I would get approximately 25.5 degrees Celsius.
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So again, the formula for converting Fahrenheit into Celsius (or vice versa) is given.
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I substituted in 78 degrees and figured this out: so, 78 degrees would be equal to approximately 25.5 degrees Celsius.
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That concludes today's lesson on Educator.com; and I will see you again for the next lesson.