Mrs. Meehan 6th Grade Algebra

Combining unlike terms I can simplify algebraic expressions

Do Now: Simplify each expression + + - = + 5 + 4 + 2 - 5 = 6

term 3X³Y coefficient variables Like terms are terms that contain the same variables raised to the same exponent. X and 3X 2Y² and -5Y² 3X²Y and ½X²Y 6 and 8 2X² and 2X r³ and t³ 3rt³ and -0.5rt³ Are Unlike Terms Are Like Terms Are Like Terms

Look at these 10 terms. Find all the like terms that can be combined:

Construct a like term that could combine with each of these terms. 7a³b² ¾mn³ 4

Combine Like Terms Example 1 7x² - 4x² = (7-4) x² = 3x² Notice that we can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Practice Simplify each expression by combing like terms. 12x + 30x = 42x 6.8y² - y² = 5.8y² -4n + 11n² -4n and 11n² are not like terms. Do not combine them -20t – 8.5t = -28.5t 1

Simplify the following expression by combining like terms: 6xy + 3x²y + xy + 6x²y + 10xy + + + + + 6xy 3x²y 1xy 6x²y 10xy 17xy 9x²y

Distributive Property n n n n n n 1 1 1 1 1 1 1 1 1 1 1 1 3(n+2) 3n + 6 3(n + 2) 3n + 6

Distributive Property What is the Distributive Property? How to use the Distributive Property in simplifying algebraic expressions? Watch video Distributive Property Basics

Example 2 Simplify 2(x+6) + 3x. Try this: 1) 6(x - 4) + 9 2) -12x - 5x + 3a + x Procedure Justification 1. 2(x+6)+3x 2. 2(x)+2(6)+3x Distributive Property 3. 2x + 12 + 3x Multiply 4. 2x + 3x + 12 Commutative Property 5. 5x + 12 Combine like terms

Solving equations I can solve equations

What is a solution of an equation? What is an equation? What is a solution of an equation? How do we find the solutions of an equation? Answer

An equation is like a balance scale. What are the rules for keeping an equation balanced? What ever you do to One side of the equal Sign must be done to The other side too Use opposite math to isolate the variable on one side of the equal sign

Motivation

2 bags + 4 blocks = 3 bags + 2 blocks How many are in one ? 2 bags + 4 blocks = 3 bags + 2 blocks - 2 bags -2 bags 4 blocks = 1 bag + 2 blocks - 2 blocks - 2 blocks 2 blocks = 1 bag

Check ? and = and Since = = 8 blocks 8 blocks

Example 1 Solve 3x – 8 = 7. Check your answer. +8 +8 3x = 15 3 3 x = 5 +8 +8 3x = 15 3 3 x = 5 Check 3x – 8 = 7 3(5) – 8 7 15 – 8 7 7 7

Example 2 Solve 4(x – 2) + 2x = 40 4x – 8 + 2x = 40 Distributive Property 6x – 8 = 40 combine like terms + 8 + 8 add 8 on both sides of the equation 6x = 48 6 6 divide 6 on both sides of the equation x = 8

Solving Algebraic Equations 1.Use the distributive property to get rid of any parenthesis 2.Combine like terms 3.Move all of the variables to one side of the equal sign (make sure it is positive!) 4.Get the variable by itself by doing opposite math to both sides of the equal sign 5.Check your answer by substituting it into the original equation

Practice -4 + 7x = 3 2a + 3 – 8a = 8 9 = 6 – (x + 2) Click here, if you need help.

I can solve systems of linear equation in two variables by elimination

Do Now: A farmer has ducks and cows. There are 8 heads and 22 feet. How many ducks and cows does he have?

Guess and Check a has __ feet ; a has __ feet 2 4 The # of Ducks The # of Cows Total # of Heads Total # of Feet 1 2 6=1*2+2*2 4 12 3 6 18 8 24 7 20 5 22

Method one: Use one variable to set up an equation. Let x = the amount of ducks, then 8-x = the amount of cows the # of ducks’ feet * the # of ducks + the # of cows’ feet * the # of cows =total # of feet 2 * x + 4 * ( 8 – x ) = 22 2x + 32 – 4x = 22 -2x + 32 = 22 -2x = -10 x = 5 the amount of cows = 8 – x = 8 – 5 = 3 So, there are 5 ducks and 3 cows.

The Idea of Elimination + + - 5 + 4 + 2 - 5 6 + - + -

Development Example 1 elimination using addition Solve x – 2y = -19 by elimination 5x + 2y = 1 Step 1 Add (1) and (2) to eliminate the y-terms. 6x = -18 Step 2 Simplify and solve for x. x = -3 Step 3 Write one of the original equations. x – 2y = -19 Step 4 Substitute -3 for x. -3 – 2y = -19 Step 5 Simplify and solve for y -2y = -16 y = 8 (1) (2)

Development Example 2 elimination using subtraction Solve 3x + 4y = 18 by elimination -2x + 4y = 8 Step 1 Subtract (1) and (2) to eliminate the y-terms. 5x = 10 Step 2 Simplify and solve for x. x = 2 Step 3 Write one of the original equations. 3x + 4y = 18 Step 4 Substitute 2 for x. 3(2) + 4y = 18 Step 5 Simplify and solve for y 6 + 4y = 18 4y = 12 y = 3 (1) (2)

Go Back to the Do Now Let d = the # of ducks and c = the # of cows In some case, we will first need to multiply one or both of the equations by a number so that one variable has opposite coefficients. This will be the new step 1. (1) * 2  2(d+c) = 8*2 2d + 2c = 16 (3) (2) – (3)  2c = 6 c = 3 cows Substitute c=3 into (1) or (2), but (1) would be easier d + c = 8  d + 3 = 8  d = 5 ducks Therefore, the farmer has 3 cows and 5 ducks.