Chapter 5: 9th Class Math Notes

Chapter 5: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about linear equations and inequalities.

This chapter requires knowledge of real numbers, including linear equations, linear inequalities, graphic inequalities, linear programming, and some techniques to solve the variables.

Chapter 5: 9th Class Math Notes

Class 9 Math Notes – Chapter 5 Summary

This summary covers key concepts and solution methods from Chapter 5 of your 9th class mathematics notes:

Solving Linear Equations

• Basic Steps:
  • Isolate the variable on one side of the equation.
  • Simplify the equation to find the value of the variable.
  • Example: 12x+30=−6 → x=−3.
• Fractional Equations:
  • Clear fractions by multiplying both sides by the least common denominator (LCD).
  • Example: x/2−3x/4=1/12​ → x=−1/3.

Solving Linear Inequalities

• Basic Steps:
  • Solve the inequality as you would an equation, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
  • Represent the solution on a number line.
  • Example: x−6≤−2 → x≤4.
• Compound Inequalities:
  • Solve each inequality separately and find the intersection of the solutions.
  • Example: −9>−16+x → x<7.

Graphical Representation of Inequalities

• Linear Inequalities in Two Variables:
  • Graph the associated linear equation (e.g., 2x+y = 6).
  • Use a test point (usually (0,0)) to determine which side of the line satisfies the inequality.
  • Shade the region that satisfies the inequality.
  • Example: 2x+y≤6→ Shade the region below the line 2x+y=6.
• Systems of Inequalities:
  • Graph each inequality on the same coordinate plane.
  • The solution region is the intersection of all shaded regions.
  • Example: x+y≥5 and −y+x≤1 → Find the overlapping shaded area.

Optimization Problems (Linear Programming)

• Objective Function:
  • A function to be maximized or minimized (e.g., f(x,y)=2x+5y).
• Constraints:
  • Inequalities that define the feasible region (e.g., 2x+y≤8, x≥0, y≥0).
• Feasible Region:
  • The area where all constraints are satisfied.
  • Corner points of the feasible region are tested to find the maximum or minimum value of the objective function.
  • Example: Maximize f(x,y)=2x+5y subject to 2x+y≤8 and x−y≤4.

Key Concepts and Formulas

• Linear Equations: ax+b=c → Solve for x.
• Linear Inequalities: ax+b<c→ Solve for xx and represent on a number line.
• Graphing Inequalities:
  • Graph the line ax+by=c.
  • Shade the region that satisfies the inequality.
• Linear Programming:
  • Objective function: f(x,y)=ax+by.
  • Constraints: ax+by≤c, x≥0, y≥0.
  • Find corner points of the feasible region to optimize the objective function.

Practice Problems

• Solving Equations: Solve equations like 2x−1/3−3x/4=5/6​.
• Solving Inequalities: Solve inequalities like 5/3x−3/4<−1/12.
• Graphing Inequalities: Graph systems of inequalities like 3x+7y≥21 and x−y≤2.
• Optimization: Maximize or minimize functions like f(x,y)=2x+3y, subject to constraints.

9th Class Math Notes Full Book