Simple Equations and Variations

Grade 11 Math Lesson - Simple Equations and Variations

Lesson Objectives

Define and solve simple linear equations.
Understand the concept of direct variation and solve related problems.
Understand the concept of inverse variation and solve related problems.
Apply knowledge of variations to real-life problems.

Lesson Introduction

Solving simple equations and understanding variations are important concepts in mathematics. Whether calculating the cost of goods or understanding how speed and time are related, linear equations and variations are applied daily.

Core Lesson Content

Definition of Simple Linear Equations: An equation of the form $ax + b = 0$ , where $a \neq 0$ .

Definition of Direct Variation: A relationship where one quantity increases as the other increases, expressed as $y = kx$ , where $k$ is the constant of variation.

Definition of Inverse Variation: A relationship where one quantity increases as the other decreases, expressed as $y = \frac{k}{x}$ , where $k$ is the constant of variation.

Worked Example

Example 1: Solve $2x + 5 = 15$

Subtract 5 from both sides:

2x = 10

Divide both sides by 2:

x = 5

Example 2: Solve $5x - 7 = 3x + 9$

Bring like terms together:

5x - 3x = 9 + 7

2x = 16

x = 8

Example 3: If $y$ varies directly as $x$ , and $y = 12$ when $x = 3$ , find the equation connecting $x$ and $y$ .

Since $y = kx$ , we have:

12 = 3k \Rightarrow k = 4

Thus, the equation is $y = 4x$

Example 4: Find $y$ when $x = 5$ using $y = 4x$

Substitute:

y = 4 \times 5 = 20

Example 5: If $y$ varies inversely as $x$ , and $y = 8$ when $x = 6$ , find the equation.

Using $y = \frac{k}{x}$ :

8 = \frac{k}{6} \Rightarrow k = 48

Equation: $y = \frac{48}{x}$

Example 6: Find $y$ when $x = 4$ using $y = \frac{48}{x}$

y = \frac{48}{4} = 12

Example 7: Solve $\frac{2x - 1}{3} = 5$

Multiply both sides by 3:

2x - 1 = 15

Add 1 to both sides:

2x = 16 \Rightarrow x = 8

Example 8: A quantity $y$ varies directly as $x$ and inversely as $z$ . If $y = 6$ when $x = 2$ and $z = 4$ , find the relation.

Using $y = k \frac{x}{z}$ :

6 = k \frac{2}{4} \Rightarrow k = 12

Thus, $y = 12 \times \frac{x}{z}$

Exercises

Solve: $x - 7 = 3$
Solve: $4(x + 2) = 20$
If $y$ varies directly as $x$ , and $y = 18$ when $x = 6$ , find $y$ when $x = 8$ .
[WAEC] Solve $5x + 2 = 3x + 14$ . [Past Question]
[NECO] Solve for $x$ if $2(x - 3) = x + 5$ . [Past Question]
If $y$ varies inversely as $x$ , and $y = 10$ when $x = 2$ , find $y$ when $x = 5$ .
Find the constant of variation if $y = 35$ when $x = 5$ in direct variation.
[JAMB] Solve $\frac{3x + 1}{2} = 4$ . [Past Question]
If $y = 5$ when $x = 3$ and $y \propto \frac{1}{x}$ , find $y$ when $x = 15$ .
A number varies directly as 5 and inversely as 2. Find the number.

Conclusion/Recap

In this lesson, we explored solving linear equations and understanding direct and inverse variations. These concepts are critical in real-world problem-solving, setting a strong foundation for quadratic equations and proportional reasoning. Up next: Quadratic Equations.