Equation Lesson - Parts, Types, and Examples

Lesson Overview

• Parts of an Equation
• Types of Equations
• Solving Different Types of Equations
• Equation vs Expression
• How to Graph an Equation
• Common Mistakes and Tips for Solving Equations
• Example of Equations

An equation is a mathematical statement showing that two things are equal. It has an equal sign (=) between two expressions.

Example: 

2x +3=7

This means that when you solve for x, the equation is balanced.

Parts of an Equation

Key Components of Algebraic Expressions

Term	Definition	Example
Variable	A symbol, like y, that represent an unknown value.	In 2y+9+15, y is the variable.
Cofficient	The number in font of a variable.	In 2y+9=15, 9 and 15 are constants
Constant	A number without a variable.	In 2y+9=15, 2 is the Coefficients
Operator	Symbols like+,-,x,* used to perform operations.	In 2y+9=15, + and = are Operations.

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Types of Equations

Type of Equation	Definition	Example
Linear Equation	The highest power of the variable is 1.	2x + 3 = 7
Quadratic Equation	The variable is raised to the power of 2.	x² - 4x + 4 = 0
Cubic Equation	The variable is raised to the power of 3.	x³ - 3x² + 2x - 1 = 0
Polynomial Equation	Variables have powers greater than 2.	x³ + 2x² + x = 0

Solving Different Types of Equations

1. Linear Equations

• Combine like terms.
• Isolate the variable.
  

Example: Solve 2x + 5 = 15

Step 1: 2x = 10 (Subtract 5 from both sides).

Step 2:  x = 5 (Divide both sides by 2).

2. Quadratic Equations

• Factorize or use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Example: Solve x² - 5x + 6 = 0

Step 1: (x - 2)(x - 3) = 0 (Factorize).

Step 2: x = 2 or x = 3.

3. Cubic Equations

• Factor or use synthetic division to simplify the cubic equation.

Example: Solve x³ - 6x² + 11x - 6 = 0

Step 1: Check for possible rational roots using the Rational Root Theorem. Try x = 1.

Step 2: Divide the cubic equation by (x - 1) using synthetic or long division.
This gives you: x² - 5x + 6 = 0.

Step 3: Factor the quadratic equation: (x - 2)(x - 3) = 0.

Step 4: Solve for x: x = 1, 2, or 3.

4. Polynomial Equations

• Factor the polynomial and solve using the Zero-Product Property.
  
  Example: Solve x⁴ - 5x² + 4 = 0
  
  Step 1: Let y = x². The equation becomes y² - 5y + 4 = 0.
  
  Step 2: Factor the quadratic equation: (y - 1)(y - 4) = 0.
  
  Step 3: Solve for y: y = 1 or y = 4.
  
  Step 4: Substitute back x² for y: x² = 1 or x² = 4.
  
  Step 5: Solve for x: x = ±1 or x = ±2.
  

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Equation vs Expression

Feature	Equation	Expression
Definition	A mathematical statement showing equality using an "=" sign.	A mathematical phrase without an "=" sign.
Contains	Variables, constants, operators, and an equal sign.	Variables, constants, and operators.
Example	2x + 5 = 10	2x + 5
Purpose	Shows a relationship between two expressions.	Represents a value or calculation.

Note:

Equations can be solved to find the value of variables.

Expressions cannot be solved but can be simplified.

How to Graph an Equation

Graphing a linear equation involves solving it and plotting the solutions on a coordinate plane. Follow these steps:

• Convert to slope-intercept form

Rewrite the equation as y = mx + b, where m is the slope and b is the y-intercept.

• Generate points

Find at least three (x, y) pairs that satisfy the equation using trial and error.

• Find intercepts
• Y-Intercept: Set x = 0 and solve for y.
• X-Intercept: Set y = 0 and solve for x.

These give the points (0, b) and (a, 0).

• Create a table

Organize the x and y values in a table for reference.

• Plot the points

Mark the points, including the intercepts, on the graph.

• Draw the line

Connect the points to form a straight line representing the equation

Example: Draw a graph of the linear equation x+3y=9.

Step 1: Rewrite the Equation in Slope-Intercept Form

The slope-intercept form is y=mx+c, where:

• m is the slope.
• c is the y-intercept.

Rearrange x+3y=9 to solve for y:

3y=−x+9(subtract x from both sides)

y=−1/3​x+3(divide by 3)

Now, the equation is y=−1/3​x+3.

Step 2: Identify Key Features

From the slope-intercept form y=−1/3​x+3:

• The slope (m) is −1/3​, which means the line decreases by 1 unit vertically for every 3 units it moves to the right.
• The y-intercept (c) is 3, meaning the line crosses the y-axis at y=3.
  

Step 3: Find Two or More Points

Choose values for x to find corresponding values for y:

Point 1: x=0 (y-intercept)y=−1/3​(0)+3=3

Point: (0,3).

Point 2: x=3y=−1/3​(3)+3=−1+3=2

Point: (3,2).

Point 3: x=6y=−1/3​(6)+3=−2+3=1

Point: (6,1).

Step 4: Plot the Points

On a graph:

• Plot (0,3) (y-intercept).
• Plot (3,2).
• Plot (6,1).
  

Step 5: Draw the Line

Use a ruler to connect the points, extending the line in both directions. This represents the equation x+3y=9.

Step 6: Verify

Optionally, test another point to confirm it lies on the line. For example:

• Let x=−3:

y=−1/3​(−3)+3=1+3=4

Point: (−3,4).

Plot (−3,4) and confirm it aligns with the line.

Graph Characteristics

• The line decreases due to the negative slope.
• The y-intercept is at (0,3).
• The x-intercept can be calculated by setting y=0 in the original equation:

x+3(0)=9⇒x=9

Thus, the x-intercept is (9,0).

By combining these insights, you have a complete graph of the equation x+3y=9.

See the values of x and y in the following table.

x

0

3

6

y

3

2

1

Fig: Graph representing the linear equation x+3y=9

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Common Mistakes and Tips for Solving Equations

Mistake	Example	Tip	Corrected Example
Not balancing both sides	2x+3=7, subtracting 3 only on one side	Apply the same operation on both sides.	2x+3−3=7−3→2x=4
Combining terms incorrectly	2x+3y=7, adding as 5xy	Combine only like terms.	2x+3y stays as is.
Ignoring negative signs	−x+5=10, solving as 𝑥+5=10	Watch for negative signs.	−x+5=10→x=−5
Misplacing fractions/powers	x/2=3, solving as 𝑥=3+2	Simplify carefully.	x/2=3→x=3×2→x=6

Example of Equations

Example 1:

Solve for the variable in the given equations:

(a) 4y + 3 = 19

(b) 2x - 5 = 11

(c) z/4 = 3

(d) a + 7 = 15

Solution:

(a) 4y + 3 = 19

4y = 19 - 3

4y = 16

y = 16 / 4

y = 4

The variable is "y," and the solution is y = 4.

(b) 2x - 5 = 11

2x = 11 + 5

2x = 16

x = 16 / 2

x = 8

The variable is "x," and the solution is x = 8.

(c) z/4 = 3

z = 3 × 4

z = 12

The variable is "z," and the solution is z = 12.

(d) a + 7 = 15

a = 15 - 7

a = 8

The variable is "a," and the solution is a = 8.

Example 2:

Emily buys two books and a bag. The total cost is $50. If the cost of the bag is $30 and each book costs the same, represent this situation as an equation and find the cost of each book.

Solution:

Let the cost of one book = $x.

The cost of the bag = $30.

The total cost = $50.

The equation is:

2x + 30 = 50

Solving:

2x = 50 - 30

2x = 20

x = 20 / 2

x = 10

The cost of each book is $10.

Example 3:

A rectangle's length is 3 times its width. The perimeter of the rectangle is 48 units. Represent this as an equation and calculate the dimensions.

Solution:

Let the width = w units and the length = 3w units.

Perimeter = 2(length + width)

The equation is:

2(3w + w) = 48

Simplifying:

2(4w) = 48

8w = 48

w = 48 / 8

w = 6

Width = 6 units, Length = 3 × 6 = 18 units.

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