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


Linear equations are an essential topic in mathematics that forms the basis for understanding more complex algebraic concepts. At its core, a linear equation is an equation that represents a straight line when graphed on a coordinate plane. They have a wide variety of applications in various fields such as engineering, physics, economics, and everyday life.

Understanding the basics of linear equations

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The general form of a linear equation in one variable is:

ax + b = 0

Here, a and b are constants, where a ≠ 0, and x is the variable. For example, 2x + 3 = 0 is a linear equation.

Straight line

The best way to understand a linear equation is to visualize it. Consider the equation y = 2x + 3 This equation can be represented as a straight line on a graph. Here's a simple visualization:

y = 2x + 3

Intercept and slope

A linear equation can be written in the form:

y = mx + c

In this equation:

  • y is the dependent variable
  • x is the independent variable
  • m is the slope of the line
  • c is the y-intercept

The slope m tells us how steep the line is, and c tells us where the line crosses the y-axis.

Solving linear equations

Solving a linear equation means finding the value of the variable that makes the equation true. Let's look at a simple example.

Example 1: Solve the equation 3x + 5 = 11.

  1. Subtract 5 from both sides to isolate the term containing the variable: 
    3x + 5 - 5 = 11 - 5
    3x = 6
  2. Divide both sides by 3 to find the value of x: 
    x = 6 / 3
    x = 2

Thus, the solution of 3x + 5 = 11 is x = 2.

Visual representation

Seeing how a linear equation forms a line can help deepen your understanding. Consider the equation y = -x + 5 If you plot it on a graph, it looks like this:

y = -x + 5

Notice that this line slopes downward from left to right, because the slope is negative -1.

Linear equations in two variables

When we expand linear equations to two variables, the general form becomes:

ax + by = c

where a, b, and c are constants, and x and y are variables.

Example 2: Solve the equation 2x + 3y = 6.

To find the solution set, we need the intersection points of the line represented by the equation. It can be solved using substitution or elimination methods, depending on what other equations are present to form the system.

Graphing linear equations in two variables

Consider plotting the linear equation x - y = 2 You can create a simple X,Y table of values to plot the points.

X | Y
----- 
0 | -2 
2 | 0 
4 | 2

When you graph these points, the line will show the equation x - y = 2.

Using slope-intercept form

Converting equations into slope-intercept form y = mx + c makes them easier to understand and graph. For example, the equation 3x - y = 3 can be transformed as follows:

y = 3x - 3

Now, it is clear that the slope m is 3 and the y-intercept is c -3.

Real life applications of linear equations

Linear equations model real-world situations where the rate of change is constant. For example:

  • Budgeting: If you earn a certain amount per hour, your total income can be represented using a linear equation.
  • Distance and Speed: Calculating travel time with constant speed involves the use of linear equations.
  • Supply and Demand: Economists use linear equations to model costs, income, and expenditures.

Example of application in the real world

Let's say you're saving money every week. If you save $50 every week, a linear equation can help you determine how much money you'll have after a certain number of weeks. Let's write this situation as a linear equation:

s = 50w

Where s is the total savings and w is the number of weeks. For example, after 10 weeks, you will have:

s = 50 * 10 = 500

You will have 500 dollars.

Conclusion

Linear equations are a powerful tool in mathematics for solving and modeling real-world problems. By understanding how to identify and manipulate their components, create graphs, and solve equations, students equip themselves with a basic skill that will be useful in more advanced mathematical concepts and practical applications. Practicing and exploring these principles will strengthen understanding and mathematical confidence.


Grade 7 → 2.2


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Linear Equations

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