Solving Equations Student Notes - MS. FORD'S MATH CLASS

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Solving Multi-Step Equations

● Variable: a letter which represents an unknown number (any letter can be used as a variable)

a x y z r v d

● Coefficient: the number in front of (touching) a variables (implies multiplication)

3c -15x𝟐

𝟑𝐤

● Constant: a number

5 8 -12 6.25 𝟓

𝟖

● Operation:+ ÷ − ×

Important Vocabulary

● Expression: a mathematical phrase made up of variables and/or constants and operations

𝟑𝒙 + 𝟓● Equation: a sentence that states that two

mathematical expressions are equal.

−𝟐𝒈 − 𝟗 = 𝟏𝟓● Inequality: a sentence that states that two

mathematical expressions are not equal.

𝟏𝟐 + 𝟓𝒚 < −𝟏𝟎

Important Vocabulary

● Term: when addition or subtractions signs separate an algebraic expression, each part is a term

𝟑𝒙 + 𝟒𝒚 − 𝟖

● Inverse: opposite

● Isolate: to be alone

Important Vocabulary

Term 1 Term 2 Term 3

What’s the point!?

We want to know the value of the unknown number.

In order to do so, we have to ISOLATEthe variable using INVERSE operations.

2x + 8 = 12

2x + 8 = 12

The inverse of adding 8 is subtracting 8

*Whatever you do to one side of the equation you must also do to the other side to keep it balanced!!

+8 and -8 cancel each other out (they make 0)

Draw a line at the equal sign to divide your equation into 2 parts.

2x =4

The inverse of multiplying by 2 is dividing by 2

Dividing 2x by 2 leaves 1 x (the variable is now isolated!)

*Whatever you do to one side of the equation you must also do to the other side to keep it balanced!!

Step 4: Check your answer

2x + 8 = 12 x = 2

2(2) + 8 = ?

4 + 8 = 12

Solve for x.

x – 5 = 6

Solve for x.

2x + 3 = 9

Solve for x.

-2x – 4 = 11

5x - 7 = -x + 35

5x - 7 = -x + 35

The inverse of subtracting 7 is adding 7

*Whatever you do to one side of the equation you must also do to the other side to keep it balanced!!

-7 and +7 cancel each other out (they make 0)

Draw a line at the equal sign to divide your equation into 2 parts.

5x = -x + 42

The inverse of subtracting x is adding x

*Whatever you do to one side of the equation you must also do to the other side to keep it balanced!!

-x and +x cancel each other out (they make 0)

The inverse of multiplying by 6 is dividing by 6

Dividing 6x by 6 leaves 1 x (the variable is now isolated!)

*Whatever you do to one side of the equation you must also do to the other side to keep it balanced!!

5x - 7 = -x + 35 x = 7

5(7) - 7 = -(7) + 35

35 – 7 = -7 + 35

28 = 28

Solve for r.

6𝑟 + 7 = 12 + 7𝑟

Solve for x.

12𝑟 − 4𝑥 = 1 − 𝑥

Solve for n.

𝑛 + 2 = −14 − n

Solve for b.

4𝑏 − 7 = 6𝑏 + 1

Solve for f.

11 − 8𝑓 = −5𝑓 + 2

● Like Terms: terms that have the same variables raised to the same exponent (or power), but have different coefficients

● Distributive Property: the process of distributing the number on the outside of the parentheses to each term on the inside.

Important Vocabulary

Like terms because each term consists of a single variable, x, and a numeric coefficient:

2x, 45x, x, 0x, -26x, -x

Like terms because they are all constants:

15, -2, 27, 9043, 0.6

Like terms because they are all y² with a coefficient: 3y², y², -y², 26y²

What are unlike terms?

The following two terms both have a single variable, but the terms are not alike since different variables are used: 17x, 17z

Each y variable in the terms below has a different exponent, therefore these are unlike terms: 15y, 19y², 31y5

Although both terms below have an x variable, only one term has the y variable, thus these are not like terms either: 19x, 14xy

5 cats + 3 cats

5a + 3a

5 apples + 3 oranges

5 cats + 3 dogs

You Try

3y + 2 + 3x – y + 5x

x + x

Solve the equation below.

Solve the equation below.

Solve the equation below.

Solve the equation below.

Distribute by multiplication:

15n and 20 are not alike and therefore cannot be combined. The answer 15n + 20 is

simplified because we do not know what the value of n is at this time and cannot

complete the multiplication part of this problem.

)43(5 n )3(5 n

)4(5

n15 20

The Distributive Property

Expressions with variables:

Simplify 5(3n + 4).

No symbol between the 5 and the parenthesis indicates a multiplication problem.

Use the Distributive Property 3 (2x – 5)

6x – 15

Distributive Property

You Try

6(2d+3)

-2(-4+5v)

Solve: 2x + 6(x + 1)

Explain how each of the below answers are wrong and why.

2x + 6x + 1

9x

Think – Pair – Share

Solve the equation below.

Solve the equation below.

Solve the equation below.

Solve the equation below.