Week-4_Linear-Equations-in-Two-Variables

H Mathematics 8

Linear Equations

Objectives

  1. Convert Linear Equations

    • Rewrite the linear equation in standard form Ax + By = C into the form y = mx + b and vice versa.

  2. Graphing Linear Equations

    • Graph a linear equation given:

      • (a) any two points

      • (b) the X- and Y-intercepts

      • (c) the slope and a point on the line.

  3. Describing the Graph

    • Describe the graph of a linear equation in terms of its intercepts and slope.

Linear Equation in Two Variables


  • A linear equation can be expressed in the standard form:Ax + By = C where A, B, and C are real numbers, and A and B are both nonzero.


  • It can also be written in slope-intercept form:y = mx + b where m represents the slope, and b is the y-intercept.

Examples of Converting Linear Equations

Example 1:

2x + y = -7

  • Solution:

    • Rearranging the equation:

    • 0 + y = -2x - 7

    • Thus, y = -2x - 7

      • Used the Subtraction Property of Equality (subtracting both sides by 2x).

Example 2:

5x - 10y = 4

  • Solution:

    • Rearranging:

    • 0 - 10y = -5x + 4

    • y = (1/2)x - 2

      • Applied Subtraction Property and Division Property of Equality.

Example 3:

x + 3y = 9

  • Solution:

    • 0 + 3y = -x + 9

    • y = -1/3x + 3

Writing Linear Equations in Standard Form

  • Any linear equation can be expressed in standard form or slope-intercept form by applying properties of equality.

Example 1:

  • y = -2x - 7

  • Rearranging:

    • 2x + y = -7

      • Used the Addition Property of Equality.

Example 2:

  • y = 8x + 1

  • Rearranging:

    • -8x + y = 1

      • Used the Subtraction Property of Equality.

Example 3:

  • y = -5/3x + 4

  • Rearranging:

    • 5x + 3y = 12

      • Applied Addition Property and the Multiplication Property of Equality.

Graphing Linear Equations in Two Variables

  • The graph of a linear equation is always a straight line.

Steps to Graph Linear Equations by Plotting Two Points:

  1. Choose a value for X

  2. Substitute to find Y

  3. Repeat to find another point

  4. Plot and connect

Example for Graphing:

  • 2x + y = 4

    • If x = -1, then y = 6

    • If x = 3, then y = -2

    • Points: (-1, 6) and (3, -2)

Table of Values:

1st Example (2x + y = 4)


  • If x = -1:y = 6


  • If x = 3:y = -2

Plotting Points Example:

  • Points: (-1, 6) and (3, -2)

  • Connect the points to graph the equation.

Graphing Linear Equations Using Intercepts:

Steps:

  1. Determine X-intercept: Set y = 0

  2. Determine Y-intercept: Set x = 0

  3. Plot both intercepts and connect.

Example for Graphing Using Intercepts:

  • 2x + y = 4

    • X-intercept: (2, 0)

    • Y-intercept: (0, 4)

Conclusion

  • Understanding how to convert, write in different forms, graph, and describe linear equations is essential in mathematics. It lays the foundation for analyzing relationships between variables.