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Solving Linear Equations

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Like Terms

Like terms contain the same variables raised to the same powers.

To combine like terms, add or subtract the numerical coefficients (as appropriate), then multiply the result by the common variable factors.

You can combine like terms by adding or subtracting them (this is not true for unlike terms).

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Combining Like Terms

6x2 + 7x2

19xy – 30xy

13xy2 – 7x2y

13x2

– 11xy

Can’t be combined (since the terms are not like terms)

Examples of Combining Terms

Terms Before Combining After Combining Terms

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Combining Like Terms

We cannot combine a chicken and a goat and create a

Chickengoat

It would be unbalanced!!!

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Combining Like Terms

Or a

Zonkey???

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Using the Distributive Property

a.) 2(x + y)

= 2x + 2y

Example: Find each product by using the distributive property to remove the parentheses.

b.) 7(x + 2y – 5z) = 7x + 14y – 35z

c.) – 4(3a – 3b – 10c) = – 12a + 12b + 40c

a.) 2(x + y)

b.) 7(x + 2y – 5z) c.) – 4(3a – 3b – 10c)

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Remember to keep it balanced by distributing to everyone

Using the Distributive Property

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Addition Property of Equality

If a, b, and c are real numbers, then

a = b and a + c = b + c

are equivalent equations.

Addition Property of Equality

z = – 16 Simplify both sides.

Example:

8 + (– 8) + z = – 8 + – 8 Add –8 to each side.

8 + z = – 8a.)

Keep i

t bala

nced

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Example:

Solving Equations

5(3 + z) – (8z + 9) = – 4z

15 + 5z – 8z – 9 = – 4z Use distributive property.

6 – 3z = – 4z Simplify left side.

6 + z = 0 Simplify both sides.

z = – 6 Simplify both sides.

6 – 3z + 4z = – 4z + 4z Add 4z to both sides.

6 + (– 6) + z = 0 +( – 6) Add –6 to both sides.

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Multiplication Property of EqualityIf a, b, and c are real numbers, then

a = b and ac = bc are equivalent equations

Multiplication Property of Equality

Example:– y = 8

y = – 8 Simplify both sides.

(– 1)(– y) = 8(– 1) Multiply both sides by –1.

Balanced

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Divide both sides by 3.3

27

3

3

z

Example:

Using Both Properties

3z – 1 = 26

3z = 27 Simplify both sides.

z = 9 Simplify both sides.

3z – 1 + 1 = 26 + 1 Add 1 to both sides.

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Solving Linear Equations

Solving Linear Equations in One Variable1) Multiply both sides by the LCD to clear the equation of

fractions if they occur.2) Use the distributive property to remove parentheses if

they occur.3) Simplify each side of the equation by combining all like

terms.4) Get all variable terms on one side and all numbers on the

other side by using the addition property of equality.5) Get the variable alone by using the multiplication

property of equality.6) Check the solution by substituting it into original

equation.