Table of Contents Inverse Operations Click on a topic to One Step - - PDF document

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www.njctl.org 2012-11-16

Solving Equations Algebra I

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Table of Contents

Inverse Operations One Step Equations Two Step Equations Multi-Step Equations Variables on Both Sides More Equations Transforming Formulas

Click on a topic to go to that section.

These sections are a repeat from 8th grade Common Core. Use with your students as necessary.

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Inverse Operations

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What is an equation? An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in 2+3=5 9-2=7

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Equations can also be used to state the equality of two expressions containing one or more variables. In real numbers we can say, for example, that for any given value of x it is true that 4x + 1 = 14 - 1 If x = 3, then 4(3) + 1 = 14 - 1 12 + 1 = 13 13 = 13

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When defining your variables, remember... Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand. While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes. Try It! Write an equation with a variable and have a classmate identify the variable and its value.

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An equation can be compared to a balanced scale. Both sides need to contain the same quantity in order for it to be "balanced".

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For example, 20 + 30 = 50 represents an equation because both sides simplify to 50. 20 + 30 = 50 50 = 50 Any of the numerical values in the equation can be represented by a variable. Examples: 20 + c = 50 x + 30 = 50 20 + 30 = y

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Why are we Solving Equations? First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to. Now, we are told what it simplifies to and we need to find the value of the variable. When solving equations, the goal is to isolate the variable on

• ne side of the equation in order to determine its value (the

value that makes the equation true).

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In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides

• f the equation.

Let's review the inverses of each operation: Addition Subtraction Multiplication Division

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There are four properties of equality that we will use to solve

• equations. They are as follows:

Addition Property If a=b, then a+c=b+c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation. Subtraction Property If a=b, then a-c=b-c for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation. Multiplication Property If a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation. Division Property If a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.

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For each equation, write the inverse operation needed to solve for the variable. a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21 c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12

move move

move

move

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Think about this... To solve c - 3 = 12 Which method is better? Why? Kendra Added 3 to each side of the equation c - 3 = 12 +3 +3 c = 15 Ted Subtracted 12 from each side, then added 15. c - 3 = 12

• 12 -12

c - 15 = 0 +15 +15 c = 15

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Think about this... In the expression To which does the "-" belong? Does it belong to the x? The 5? Both? The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable. So:

Touch to reveal answer

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One Step Equations

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To solve equations, you must work backwards through the

• rder of operations to find the value of the variable.

Remember to use inverse operations in order to isolate the variable on one side of the equation. Whatever you do to one side of an equation, you MUST do to the other side!

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Examples: y + 9 = 16

• 9 -9 The inverse of adding 9 is subtracting 9

y = 7 6m = 72 6 6 The inverse of multiplying by 6 is dividing by 6 m = 12 Remember - whatever you do to one side of an equation, you MUST do to the other!!!

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x - 8 = -2 +8 +8 x = 6 x + 2 = -14

• 2 -2

x = -16 2 = x - 6 +6 +6 8 = x 7 = x + 3

• 3 -3

4 = x 15 = x + 17

• 17 -17
• 2 = x

x + 5 = 3

• 5 -5

x = -2

One Step Equations

Solve each equation then click the box to see work & solution.

click to show inverse operation click to show inverse operation click to show inverse operation click to show inverse operation click to show inverse operation click to show inverse operation

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One Step Equations 3x = 15 3 3 x = 5

• 4x = -12
• 4 -4

x = 3

• 25 = 5x

5 5

• 5 = x

click to show inverse operation click to show inverse operation click to show inverse operation

x 2 x = 20 = 10 (2) (2)

x

• 6

x = -216 = 36

click to show inverse operation

(-6) (-6)

click to show inverse operation

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Two-Step Equations

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Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable. This means that you undo in the opposite order (PEMDAS): 1st: Addition & Subtraction 2nd: Multiplication & Division 3rd: Exponents 4th: Parentheses Whatever you do to one side of an equation, you MUST do to the other side!

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Examples: 3x + 4 = 10

• 4 - 4 Undo addition first

3x = 6 3 3 Undo multiplication second x = 2

• 4y - 11 = -23

+ 11 +11 Undo subtraction first

• 4y = -12
• 4 -4 Undo multiplication second

y = 3 Remember - whatever you do to one side

• f an equation, you MUST do to the other!!!

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6-7x = 83

• 6 -6
• 7x = 77
• 7 -7

x = -11 3x + 10 = 46

• 10 -10

3x = 36 3 3 x = 12

• 4x - 3 = 25

+3 +3

• 4x = 28
• 4 -4

x = -7

• 2x + 3 = -1
• 3 -3
• 2x = -4
• 2 -2

x = 2 9 + 2x = 23

• 9 -9

2x = 14 2 2 x = 7 8 - 2x = -8

• 8 -8
• 2x = -16
• 2 -2

x = 8

Two Step Equations

Solve each equation then click the box to see work & solution.

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Multi-Step Equations

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Slide 50 / 116 Steps for Solving Multiple Step Equations

As equations become more complex, you should:

• 1. Simplify each side of the equation.

(Combining like terms and the distributive property)

• 2. Use inverse operations to solve the equation.

Remember, whatever you do to one side of an equation, you MUST do to the other side!

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Examples:

• 15 = -2x - 9 + 4x
• 15 = 2x - 9

Combine Like Terms +9 +9 Undo Subtraction first

• 6 = 2x

2 2 Undo Multiplication second

• 3 = x

7x - 3x - 8 = 24 4x - 8 = 24 Combine Like Terms + 8 +8 Undo Subtraction first 4x = 32 4 4 Undo Multiplication second x = 8

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Now try an example. Each term is infinitely cloned so you can pull them down as you solve.

• 7x + 3 + 6x = -6

x = 9

answer

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Now try another example. Each term is infinitely cloned so you can pull them down as you solve. 6x

• 5

+ x = 44

x = -9

answer

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Always check to see that both sides of the equation are simplified before you begin solving the equation. Sometimes, you need to use the distributive property in

• rder to simplify part of the equation.

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For all real numbers a, b, c a(b + c) = ab + ac a(b - c) = ab - ac

Distributive Property Slide 56 / 116

Examples 5(20 + 6) = 5(20) + 5(6) 9(30 - 2) = 9(30) - 9(2) 3(5 + 2x) = 3(5) + 3(2x)

• 2(4x - 7) = -2(4x) - (-2)(7)

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Examples: 5(1 + 6x) = 185 5 + 30x = 185 Distribute the 5 on the left side

• 5 -5

Undo addition first 30x = 180 30 30 Undo multiplication second x = 6 2x + 6(x - 3) = 14 2x + 6x - 18 = 14 Distribute the 6 through (x - 3) 8x - 18 = 14 Combine Like Terms +18 +18 Undo subtraction 8x = 32 8 8 Undo multiplication x = 4

Move to reveal answer

Move to reveal answer

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5 ( -2 + 7x ) = 95

Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down) x = 3

answer

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6 ( -2x + 9 ) = 102

Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down) x = -4

answer

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Variables on Both Sides

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Remember...

• 1. Simplify both sides of the equation.
• 2. Collect the variable terms on one side of the equation.

(Add or subtract one of the terms from both sides of the equation)

• 3. Solve the equation.

Remember, whatever you do to one side of an equation, you MUST do to the other side!

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Example: 4x + 8 = 2x + 26

• 2x -2x

Subtract 2x from both sides 2x + 8 = 26

• 8 -8 Undo Addition

2x = 18 2 2 Undo Multiplication x = 9 What if you did it a little differently? 4x + 8 = 2x + 26

• 4x -4x

Subtract 4x from both sides 8 = -2x + 26

• 26 - 26

Undo Addition

• 18 = -2x
• 2 -2 Undo Multiplication

9 = x Recommendation: Cancel the smaller amount of the variable!

Slide down to reveal steps Slide down to reveal steps

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Example: 6r - 5 = 7r + 7 - 2r 6r - 5 = 5r + 7 Simplify Each Side of Equation

• 5r -5r Subtract 5r from both sides (smaller than 6r)

r - 5 = 7 + 5 +5 Undo Subtraction r = 12

Slide down to reveal steps

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Try these: 6x - 2 = x + 13 4(x + 1) = 2x -2 5t - 8 = 9t - 10

• x -x

4x + 4 = 2x -2

• 5t -5t

5x - 2 = 13

• 2x -2x
• 8 = 4t - 10

+ 2 +2 2x + 4 = -2 +10 +10 5x = 15

• 4 -4

2 = 4t 5 5 2x = -6 4 4 x = 3 2 2 = t x = -3

1 2

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Sometimes, you get an interesting answer. What do you think about this? What is the value of x? 3x - 1 = 3x + 1 Since the equation is false, there is " no solution"! No value will make this equation true.

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Sometimes, you get an interesting answer. What do you think about this? What is the value of x? 3(x - 1) = 3x - 3 Since the equation is true, there are infinitely many solutions! The equation is called an identity. Any value will make this equation true.

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Try these: 4y = 2(y + 1) + 3(y - 1) 14 - (2x + 5) = -2x + 9 9m - 8 = 9m + 4 4y = 2y + 2 + 3y - 3 14 - 2x - 5 = -2x + 9 - 9m - 9m 4y = 5y - 1 9 - 2x = -2x + 9

• 8 = 4
• 5y -5y

+2x +2x No Solution

• y = -1

9 = 9 y = 1 Identity

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Mary's distance (rate time) equals Jocelyn's distance (rate time) Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?

Define

t = Mary's time in hours t + 0.25 = Jocelyn's time in hours

Relate Write

12t = 9(t+0.25)

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12t = 9(t + 0.25) 12t = 9t + 2.25

• 9t -9t

3t = 2.25 3 3 t = 0.75 It took Mary 0.75h, or 45 min, to get home. Step 1 - distribute the 9 inside the parenthesis

(pull)

Step 2 - subtract 9t from both sides

(pull)

Step 3 - divide both sides by 3

(pull)

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More Equations

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Remember...

• 1. Simplify each side of the equation.
• 2. Collect the variable terms on one side of the equation.

(Add or subtract one of the terms from both sides of the equation)

• 3. Solve the equation.

(Undo addition and subtraction first, multiplication and division second)

Remember, whatever you do to one side of an equation, you MUST do to the other side!

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5 3

Examples: x = 6 x = 6 Multiply both sides by the reciprocal x = x = 10

2x - 3 = + x

• x - x Subtract x from both sides

x - 3 = +3 +3 Undo Subtraction x =

3 5 3 5 5 3 30 3

• 14

5

• 14

5 1 5

Click to reveal steps Click to reveal steps

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There is more than one way to solve an equation with distribution.

Multiply by the reciprocal Multiply by the LCM

(-3 + 3x) =

3 5 72 5

(-3 + 3x) =

3 5 72 5

(-3 + 3x) =

3 5 72 5

(-3 + 3x) =

3 5 72 5 5 3 5 3

• 3 + 3x = 24

+3 +3 3x = 27 3 3 x = 9 (-3 + 3x) =

3 5 72 5

5 5 3(-3 + 3x) = 72

• 9 + 9x = 72

+9 +9 9x = 81 9 9 x = 9

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Transforming Formulas

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Formulas show relationships between two or more variables. You can transform a formula to describe one quantity in terms of the others by following the same steps as solving an equation.

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Example: Transform the formula d = r t to find a formula for time in terms of distance and rate. What does "time in terms of distance and rate" mean? d = r t r r = t d r Divide both sides by r

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Examples V = l wh Solve for w V = w l h P = 2l + 2w Solve for l

• 2w
• 2w

P - 2w = 2l 2 2 P - 2w = l 2

Slide to reveal steps

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Example: To convert Fahrenheit temperature to Celsius, you use the formula: C = (F - 32) Transform this formula to find Fahrenheit temperature in terms of Celsius temperature. (see next page) 5 9

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C = (F - 32) C = F - + + C + = F C + 32 = F

5 9 5 9 160 9 160 9 160 9 5 9 160 9 9 5 9 5 (

)

9 5

Solve the formula for F

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Transform the formula for area of a circle to find radius when given Area. A = r2 = r

2

A

= r

A Slide to reveal answer

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Solve the equation for the given variable. m p n q m p n q mq p n = for p = (q) = (q) 2(t + r) = 5 for t 2(t + r) = 5 2 2 t + r =

• r - r

t = - r 5 2 5 2

Move to reveal steps Move to reveal steps

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