Solving by Elimination MCQs Quiz | Class 10

This quiz covers Class X Mathematics (Code 041) from Unit II: Algebra, specifically focusing on Solving by Elimination, including its method steps and applications. Attempt all questions, submit, and download your personalized answer PDF.

Understanding the Elimination Method for Linear Equations

The elimination method is a powerful algebraic technique used to solve systems of linear equations. Its primary goal is to eliminate one of the variables by making its coefficients equal in magnitude but opposite in sign (or just equal in magnitude) in both equations, and then adding or subtracting the equations. This results in a single linear equation with only one variable, which is then easy to solve.

Key Steps of the Elimination Method:

  1. Standard Form: Ensure both equations are in the standard form Ax + By = C.
  2. Equal Coefficients: Choose one variable to eliminate (say, x or y). Multiply one or both equations by suitable non-zero constants so that the absolute values of the coefficients of the chosen variable become equal in both equations.
  3. Eliminate the Variable:
    • If the equal coefficients have opposite signs, add the two equations.
    • If the equal coefficients have the same sign, subtract one equation from the other.
    This step eliminates one variable, leaving a linear equation in the other variable.
  4. Solve for Remaining Variable: Solve the resulting single-variable equation to find the value of that variable.
  5. Substitute Back: Substitute the value found in step 4 into any one of the original equations (or modified equations) to find the value of the second variable.
  6. Verify Solution: Check your solution by substituting both variable values into both original equations. Both equations should be satisfied.

When to Use Elimination vs. Substitution:

While both methods can solve any system of linear equations, elimination is often preferred when:

  • No variable has a coefficient of 1 or -1, making substitution involve fractions early.
  • The coefficients of one variable are already opposites or easily made opposites.
  • The equations are already in the standard form Ax + By = C.

Applications of Systems of Linear Equations (Solvable by Elimination):

Systems of linear equations are widely used to model real-world scenarios and solve problems across various fields. Here are a few common application types:

Application Type Description Example
Age Problems Involving ages of two people or a person’s age at different times. The sum of the ages of a father and his son is 60. The father’s age is three times the son’s age. Find their ages.
Money/Cost Problems Dealing with quantities and their prices, or distribution of money. A shop sells pens for Rs 5 each and notebooks for Rs 12 each. If a student buys a total of 7 items and spends Rs 54, how many pens and notebooks did they buy?
Number Problems Relating two unknown numbers based on given conditions (sum, difference, product, ratios). The sum of two numbers is 25, and their difference is 7. Find the numbers.
Geometry Problems Finding dimensions (length, width) of shapes based on perimeter or area relations. The perimeter of a rectangle is 40 cm. The length is 4 cm more than the width. Find the length and width.

Quick Revision Points:

  • The goal is to eliminate one variable to solve for the other.
  • Multiply equations to make coefficients of one variable equal or opposite.
  • Add if signs are opposite, subtract if signs are same.
  • Always substitute the first found value back into an original equation.
  • Don’t forget to check your solution!

Practice Questions:

Try solving these extra questions using the elimination method:

  1. Solve: 3x + 2y = 11 and 2x – 3y = -10
  2. Find two numbers whose sum is 36 and whose difference is 12.
  3. The cost of 5 apples and 3 bananas is Rs 60. The cost of 2 apples and 4 bananas is Rs 46. Find the cost of one apple and one banana.
  4. A boat travels 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can travel 40 km upstream and 55 km downstream. Determine the speed of the stream and the speed of the boat in still water.
  5. For what value of k does the system of equations 2x + ky = 1 and 3x – 5y = 7 have no solution?