Lesson Note
INTRODUCTION
Simple equations and variations form the backbone of algebraic thinking in senior secondary mathematics. They allow students to model real‑world relationships, to manipulate symbols, and to predict outcomes. Understanding how a quantity changes with another is essential for solving problems in physics, economics, and everyday life.
In this lesson we will explore the structure of simple equations, learn how to solve them, and then extend the concepts to direct, inverse, joint and combined variations. By the end of the unit students will be able to set up and solve equations that describe how one variable depends on one or more others.
MAIN CONTENT
Simple Equations
Simple equations are algebraic statements that contain one or more variables and are set equal to a constant or another expression. The goal is to isolate the variable and determine its value.
Example
Step 1: Add $5$ to both sides.
Step 2: Divide by $3$.
Final answer: $x = 7$.
Practice Questions
Practice Question 1: Solve $2x + 7 = 19$.
Answer: $x = 6$.
Practice Question 2: Solve $5 - 4y = 1$.
Answer: $y = 1$.
Direct Variation
When two quantities vary directly, one is a constant multiple of the other. The relationship is expressed as $y = kx$, where $k$ is the constant of variation.
Example
When $x = 4$, $y = 12$.
Find $k$ and then $y$ when $x = 7$.
Step 1: Substitute $x = 4$, $y = 12$.
Step 2: Solve for $k$.
Step 3: Substitute $x = 7$ and $k$ to find $y$.
Final answer: $k = 3$, $y = 21$.
Practice Questions
Practice Question 1: If $y$ varies directly with $x$ and $y = 10$ when $x = 5$, find $k$ and $y$ when $x = 12$.
Answer: $k = 2$, $y = 24$.
Practice Question 2: Find $x$ when $y = 18$ and $k = 3$.
Answer: $x = 6$.
Inverse Variation
When one quantity varies inversely with another, their product is constant: $y = \dfrac{k}{x}$.
Example
When $y = 8$ and $x = 2$, find $k$ and then $y$ when $x = 5$.
Step 1: Substitute $x = 2$, $y = 8$.
Step 2: Solve for $k$.
Step 3: Substitute $x = 5$ and $k$ to find $y$.
Final answer: $k = 16$, $y = 3.2$.
Practice Questions
Practice Question 1: If $y$ varies inversely with $x$ and $y = 15$ when $x = 3$, find $k$ and $y$ when $x = 5$.
Answer: $k = 45$, $y = 9$.
Practice Question 2: Find $x$ when $y = 12$ and $k = 48$.
Answer: $x = 4$.
Joint Variation
When a quantity varies jointly with two or more others, the relationship is $y = kxz$ (for two variables) or $y = kx_1x_2\dots x_n$.
Example
When $y = 48$, $x = 3$, $z = 4$, find $k$ and then $y$ when $x = 6$, $z = 2$.
Step 1: Substitute $x = 3$, $z = 4$, $y = 48$.
Step 2: Solve for $k$.
Step 3: Substitute $x = 6$, $z = 2$ and $k$ to find $y$.
Final answer: $k = 4$, $y = 48$.
Practice Questions
Practice Question 1: If $y$ varies jointly with $x$ and $z$ and $y = 30$ when $x = 2$, $z = 5$, find $k$ and $y$ when $x = 4$, $z = 3$.
Answer: $k = 3$, $y = 36$.
Practice Question 2: Find $x$ when $y = 60$, $k = 5$, $z = 3$.
Answer: $x = 4$.
Combining Variations
A variable may vary directly with one quantity and inversely with another: $y = \dfrac{kx^2}{z}$.
Example
When $y = 20$, $x = 2$, $z = 5$, find $k$ and then $y$ when $x = 4$, $z = 10$.
Step 1: Substitute $x = 2$, $z = 5$, $y = 20$.
Step 2: Solve for $k$.
Step 3: Substitute $x = 4$, $z = 10$ and $k$ to find $y$.
Final answer: $k = 5$, $y = 8$.
Practice Questions
Practice Question 1: If $y$ varies directly with $x^3$ and inversely with $z$, and $y = 54$ when $x = 3$, $z = 2$, find $k$ and $y$ when $x = 2$, $z = 4$.
Answer: $k = 6$, $y = 12$.
Practice Question 2: Find $x$ when $y = 32$, $k = 8$, $z = 2$.
Answer: $x = 2$.
Applications of Variation
Variations are used to model physical laws. For example, speed $v$ varies directly with distance $d$ and inversely with time $t$: $v = \dfrac{k d}{t}$.
Example
When $v = 60$ km/h, $d = 30$ km, $t = 0.5$ h, find $k$ and then $v$ when $d = 45$ km, $t = 0.75$ h.
Step 1: Substitute $v = 60$, $d = 30$, $t = 0.5$.
Step 2: Solve for $k$.
Step 3: Substitute $d = 45$, $t = 0.75$ and $k$ to find $v$.
Final answer: $k = 1$, $v = 60$ km/h.
Practice Questions
Practice Question 1: If $v$ varies directly with $d$ and inversely with $t$, and $v = 80$ km/h when $d = 40$ km, $t = 0.5$ h, find $k$ and $v$ when $d = 60$ km, $t = 0.75$ h.
Answer: $k = 4$, $v = 80$ km/h.
Practice Question 2: Find $t$ when $v = 90$ km/h, $k = 2$, $d = 30$ km.
Answer: $t = 0.666...$ h.
Systems of Variation
Sometimes two variables vary jointly with a third. For instance, $x = k_1 z$, $y = k_2 z$.
Example
When $x = 12$ and $y = 18$ for $z = 3$, find $k_1$, $k_2$, and then $x$ and $y$ for $z = 6$.
Step 1: Solve for $k_1$ and $k_2$ using $z = 3$.
Step 2: Substitute $z = 6$ to find $x$ and $y$.
Final answer: $k_1 = 4$, $k_2 = 6$, $x = 24$, $y = 36$.
Practice Questions
Practice Question 1: If $x = 8$ when $z = 2$, and $y = 30$ when $z = 5$, find $k_1$, $k_2$, and $x$, $y$ when $z = 4$.
Answer: $k_1 = 4$, $k_2 = 6$, $x = 16$, $y = 24$.
Practice Question 2: Find $z$ when $x = 20$, $k_1 = 5$.
Answer: $z = 4$.
SUMMARY
The study of simple equations and variations equips students with the tools to translate real‑world relationships into algebraic form. Solving linear equations builds fluency with algebraic manipulation. Direct, inverse, joint and combined variations reveal how quantities depend on one another, and the constant of variation is key to modelling. By mastering these concepts, students can tackle problems in physics, economics, and everyday life with confidence.
EVALUATION
Solve for $x$: $4x + 5 = 21$
Solve for $y$: $y = \frac{12}{x}$ when $x = 3$
Find $k$: $y = kx$ when $y = 15$ and $x = 5$
Determine $z$: $x = 8z$ when $x = 32$
Compute $y$: $y = \frac{5x^2}{z}$ when $x = 4$ and $z = 10$