Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Algebra → Linear Equations ↓


Applications of Linear Equations


Linear equations are an essential part of algebra and mathematics in general. They are used to describe relationships where the rate of change is constant. When studying linear equations, we often encounter situations where we have to find the missing value that makes the equation true. In this comprehensive look, we'll explore how linear equations apply to real-world problems, and how you can use them as powerful tools in everyday life.

What is a linear equation?

A linear equation is an equation that forms a straight line when graphed. A linear equation can be written as:

y = mx + b

Where:

  • y is the dependent variable.
  • m is the slope of the line, or how steep the line is.
  • x is the independent variable.
  • b is the y-intercept, or where the line intersects the y-axis.

This is known as the slope-intercept form of a linear equation. The main feature of a linear equation is that the change between the variables remains constant.

Visualization of Linear Equations

Let's imagine a simple linear equation: y = 2x + 3.

X y = 2x + 3

In the above graph, you can see a line that represents the equation y = 2x + 3, which forms a straight line.

Applications in real life

Linear equations are used in a variety of fields. Let's discuss some of the applications you may encounter:

Finance

Imagine you are given the task of predicting the monthly profit of a small business. If you know that the business earns $200 from each product sold, and has a fixed cost of $500 every month, you can create a linear equation to calculate the profit:

Profit = 200x - 500

where x is the number of products sold. Using this equation, you can estimate how profits will change if sales increase or decrease.

Distance issues

Suppose you are running at a certain speed and want to calculate how much distance you will cover in a certain time. If you run at a speed of 6 mph, the distance you have run can be calculated with this linear equation:

Distance = 6t

Where t is the time in hours. This equation helps you plan your race based on the time you have available.

Time (hours) distance 6 mph

Shopping budget

You're planning a party and you need to buy chairs and tables. Each chair costs $15, and each table costs $50. If you have $200 to spend, you can use a linear equation to express this budget:

15c + 50t = 200

Where c is the number of chairs and t is the number of tables. This equation allows you to plan different combinations of chairs and tables according to your budget.

Solving linear equations

Solving linear equations involves finding the value of the variable that makes the equation true. Some techniques for solving linear equations are as follows:

Example: Simple equation

Find the value of x in the equation:

2x + 4 = 12

Step 1: Subtract 4 from both sides to isolate the term containing x.

2x + 4 - 4 = 12 - 4

This makes it simpler:

2x = 8

Step 2: Divide by 2 to solve for x.

x = 8 / 2

Simplified: x = 4

Example: Equation with fractions

Find the value of y in the equation:

3/4y - 5 = 10

Step 1: Add 5 to both sides.

3/4y - 5 + 5 = 10 + 5

This makes it simpler:

3/4y = 15

Step 2: Multiply both sides by 4/3 to solve for y.

y = (15) * (4/3)

Simplified: y = 20

Interpretation solutions

When you solve a linear equation, the solution should always make sense in the context of the problem. For example, if you are looking for the number of chairs you can buy within your budget, make sure the number is reasonable and non-negative.

Complex situations

Sometimes linear equations are not immediately obvious, and you may need to manipulate the equation to identify a linear relationship. Understanding the components of an equation allows you to construct or decompose relationships in more complex contexts, such as physics or engineering.

Practice problems

  • Consider a phone plan that charges a base fee of $10, plus $0.05 per minute of calls. If m is the number of minutes used, write a linear equation for the total cost C

  • An object travels at a speed of 50 meters per second. How much distance does it cover in t seconds? Draw a linear equation to represent this scenario.

  • If the width of a rectangle is three times its height and its perimeter is 64 units, find the dimensions of the rectangle using a linear equation.

You now have a better understanding of linear equations, their applications, and how to solve them. Linear equations are not just a subject of mathematics, but a valuable tool for simplifying and solving real-world problems.


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Applications of Linear Equations

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