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Equations,

Solutions,

Solution Sets Prepared by:

Engr. Sandra Enn Bahinting

Equation

An equation is a statement denoting the equality of two algebraic expressions.

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Linear equations

A linear equation in one variable x is an equation

that can be written in the form

ax + b = 0,

where a and b are real numbers and a ≠ 0

Example : 2x+3 = 0 x+8 = 0

3x=7 2x+5 =9-5x

3+5(x-1) =-7 + x

Solving a Linear Equation in General

Example 2: Solve for the variable. x - 5 = 2.

x - 5 = 2

x - 5 + 5 = 2 + 5

x = 7

Solve for the variable. y + 4 = -7.

y + 4 = -7

y + 4 - 4 = -7 - 4

y = -11

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Example 4: Solve for the variable.

x/2 = 5.

Example 5: Solve for the variable.

5x = 7.

Strategy for Solving a Linear Equation

Step 1: Simplify each side, if needed.

This would involve things like removing ( ), removing fractions, adding like terms, etc.

To remove ( ): Just use the distributive property found in Tutorial 5: Properties of Real Numbers.

To remove fractions: Since fractions are another way to write division, and the inverse of divide is to

multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

Step 2: Use Add./Sub. Properties to move the variable term to one side and

all other terms to the other side.

Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.

Step 4: Check your answer.

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Solve the following variables:

2(x + 5) - 7 = 3(x - 2)

Literal Equations

Equations with multiple variables where you are asked

to solve for just one of the variables.

Solve the following:

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SeatWork

Quadratic Equations Definition A quadratic equation in x is any equation that may be written in the form

𝒂𝒙𝟐+ bx + c = 0,

where a, b, and c are coefficients and a ≠ 0.

Examples 𝑥2 + 2x = 4 (2 + x)(3 – x) = 0 𝑥2 - 3 = 0 3 𝑥2 –

𝟐

𝒙+ 4 = 0 not a quadratic

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Solving Quadratic Equations Method 1 - Factoring

Here are the steps to solve a quadratic by factoring:

1. Write your equation in the form ax2 + bx + c = 0 by applying the Distributive Property, Combine Like Terms, and apply the Addition Property of Equality to move terms to one side of =.

2. Factor your equation by using the Distributive Property and the appropriate factoring technique. Note: Any type of factoring relies on the Distributive Property.

3. Let each factor = 0 and solve. This is possible because of the Zero Product Law.

Solve the following by factoring

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Solving Quadratic Equations – Method 2 – Extracting Square Roots Extracting Square Roots allows you to

rewrite 𝑥2= k as x = ±√k, where k is

some real number

Simply

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Solve by square root method

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Solving Quadratic Equations – Method 3 – Completing the Square

Solve the following:

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Solving Quadratic Equations – Method 4 – Using The Quadratic Formula Solving a quadratic equation that is in the form

𝒂𝒙𝟐 + bx + c = 0 only involves plugging a, b,

and c into the formula

the expression (𝑏2– 4ac), denoted by D, is called Discriminant,

because it determines the number of solutions or nature of roots of

a quadratic equation.

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Summary

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Solve the following:

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ASSIGNMENT (1 whole)

Square root property

Completing the square

Quadratic equation

factoring

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