MM212 Unit 2 Seminar Agenda Simplifying Algebraic Expressions Solving Linear Equations and Formulas

Philosophy Reminder from Unit 1 I do not care if you have never been good at math. I do not care if you do not like math. I do not care if you stopped taking math in 3rd grade. None of that matters to me! What does matter to me is this … #1. You give me a chance to help you #2. You maintain a POSITIVE ATTITUDE so you give yourself a chance to be successful #3. We work TOGETHER as a TEAM so we will ALL be successful. #4. NO ONE QUITS OR DISAPPEARS!

Philosophy Reminder from Unit 1 In support of our POSITIVE ATTITUDE, I have erected a NO NEGATIVE ZONE in our class This means no posts in the DB or SEMINARS about you not being good at math, not liking math, or anything negative. This things are not conducive to our learning environment and will distract us from our goal of being successful!

Simplifying Algebraic Expressions Regardless of which unit we are in … SIMPLIFYING ALGEBRAIC EXPRESSIONS will always work the same way. What it boils down to is –Clearing grouping symbols –Combining like terms –Reducing things that can be reduced

Clearing Grouping Symbols Most common method … is the Distributive Property (page 94 of the text) This is a SHARING property … we will share with EVERYTHING inside the grouping symbol – yes even the numbers we do not like! This property has to do with MULTIPLICATION!!!

6(x – 7) -13(2x – 6)

LIKE TERMS When we are combining items (adding or subtracting), we can only combine like terms. Like terms have the EXACT same variables raised to the EXACT same power. Like terms are combined by adding or subtracting the numerical coefficients AND keeping the SAME variables with the SAME exponents Plain boring numbers (those with no letters attached) … whether they are fractions, decimals, negative, whatever … call always be combined!

EXAMPLES 3 oranges + 5 oranges = 8 oranges 3x + 5x = 7x 3x 2 + 5x 2 = 8x 2 3x + 5y …. Does not combine … not like terms 3x 2 + 5y 2 …. Does not combine … not like terms 3x 2 + 5x 3 …. Does not combine … not like terms Just like 3 oranges + 5 cars does not equal 8 orange - cars!

Simplify: 3(x 2 - 4) - 4(2x 2 + 4)

Properties of Real Numbers The ADDITIVE INVERSE of a number is its OPPOSITE. ALL numbers have additive inverses. Two numbers are additive inverses if their sum is equal to zero. Inverse Property of Addition Anything plus its opposite is 0 a + (-a) = 0 and –a + a = 0

Solving Equations There are FOUR BASIC ideas … and they all work together to help us!  Solving an equation …. means to rearrange an equation to get the UNKNOWN all alone.  To get the variable all alone …. we are UNDOING all the arithmetic attached to the UNKNOWN.  To UNDO all the arithmetic … we will use the OPPOSITE arithmetic.  We can do whatever we want to an equation as long as we apply the exact same arithmetic TO BOTH SIDES OF THE EQUAL SYMBOL! These are our four guiding principles! Do NOT deviate from them!

Steps For Solving Equations STEP1: Clear the grouping symbols using the distribution property. STEP2: Clear the fractions by multiplying EVERY term by a common denominator (it does NOT have to be the least common denominator). STEP3: Move all the variables to one side of the equal symbol using the addition or subtraction property of equality. STEP4: Move all the plain boring numbers to the other side of the equal symbol using the addition or subtraction property of equality. STEP5: Isolate the variable using the multiplication or division property of equality. STEP6: Substitute your solution into the ORIGINAL equation to see if a true statement results. I use these steps … in this order … EVERYTIME! Doing them in this order REDUCES the number of arithmetic errors made!

Example 1: X + 3 = 8

Example 2. X – 17 = 28

Example = -60z

Example = x/8

Example x =

Example x =

Example 7. 3x + 4 = 2x + 9

Example 8. -9x x = 11x - 8

Example 9. 4(x + 6) = 84

Example 10 -5(x + 4) = 2(x – 4)

Example 11. Watch the technique I use to clear the denominators…this is the best way to do it…one step at a time…showing each step. (2/3) x = 4/5

Example 12. Watch the technique I use to clear the denominators…this is the best way to do it…one step at a time…showing each step. ½ x + 5/4 = 7/4

Example 13. Watch the technique I use to clear the denominators…this is the best way to do it…one step at a time…showing each step. 3 + (3/5) x = -6 + (1/8) x

Example 14. Watch the technique I use to clear the denominators…this is the best way to do it…one step at a time…showing each step. (2z +3) ∕ 3 +(3z – 4) ∕ 6 = (z – 2) ∕ 2

Example 15. Watch the technique I use to clear the denominators…this is the best way to do it…one step at a time…showing each step. (x+1) ∕ 3 – (x – 1) ∕ 6 = 1/6

Special Cases Conditional: a letter equals a number (like the examples 1 – 15) Identities: Sometimes one can work through a whole problem, and only numbers are left, no numbers. One of two things is true: a. Left with no letters, but numbers make sense, your answer would be interpreted as ALL REAL NUMBERS. Example: if very last step is 0 = 0 or = or 14/3 = 14/3 [one number = to the same number]. The solution is ALL REAL NUMBERS.

Special Cases (Identities) cont. Identities (cont.) b. Left with no letters, and numbers do not make sense, your answer would be interpreted as NO SOLUTION. Example: if very last step is 0 = -13 or =1.205 or 32/3 = ½ [one number = to another number]. The solution is NO SOLUTION. Equations of this type are called CONTRADICTIONS

Example 1: Solve 2(x + 1) = 2(x + 3)

Example 2: Solve: 8x + 3(2 – x) = 5x + 6

Solving a Formula for a Specified Variable Same or similar steps as solving an equation for x. Goal is to isolate the variable of interest…undo all the arithmetic attached to the variable of interest…in other words rearrange the equation to get the variable all alone. Step 1: Clear the grouping symbols using the distribution property. Step 2: Clear the fractions by multiplying EVERY term by a common denominator (it does NOT have to be the LCM). Step 3: Move all the variables ATTACHED TO OUR VARIABLE OF INTEREST BY ADDITION OR SUBTRACTION to the OTHER side of the equal symbol using the addition or subtraction of equality. Step 4: Isolate the variable using the multiplication or division property of equality. [When I perform this type of arithmetic after I write the problem on my paper, I put a circle or a square around the letter I am trying to isolate. This gives me focus and provides a constant reminder as to what my goal is.]

Example 1: Solve s = ½ g t 2 + v t for “g” [variable of interest]

Example 2: Solve p = 2L + 2W for “W”