Pair of Linear Equations - Class 10
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Pair of Linear Equations - Class 10

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Questions and Answers

What does the equation $3x + 4y = 20$ represent in Akhila's scenario?

  • Total number of Hoopla games played
  • Total amount of money earned from games
  • Total number of rides Akhila took
  • Total cost of rides and games (correct)

    • A pair of linear equations that has a solution is referred to as an inconsistent pair.

    False

    If Akhila had 2 rides, how many times did she play Hoopla?

    1

    A pair of linear equations which has no solution is called an _________ pair of linear equations.

    <p>inconsistent</p> Signup and view all the answers

    Match the terms with their definitions:

    <p>Consistent Pair = Has at least one solution Inconsistent Pair = Has no solutions Dependent Pair = Has infinitely many solutions Graphical Method = Method used to visualize solutions of equations</p> Signup and view all the answers

    If Akhila spent all her money, which equation describes her total spending?

    <p>$3x + 4y = 20$</p> Signup and view all the answers

    Two equations that represent the same line are considered independent.

    <p>False</p> Signup and view all the answers

    If the number of rides Akhila took is represented by x, then the number of times she played Hoopla is represented by _______.

    <p>y</p> Signup and view all the answers

    What is the solution to the system of equations 2x + 3y = 11 and 2x – 4y = –24?

    <p>(2, 2)</p> Signup and view all the answers

    The pair of linear equations for ‘The difference between two numbers is 26 and one number is three times the other’ can be represented as x - y = 26 and x = 3y.

    <p>True</p> Signup and view all the answers

    What is the value of 'm' if y = mx + 3 and the point (4, 11) lies on this line?

    <p>2</p> Signup and view all the answers

    The larger of two supplementary angles exceeds the smaller by ____ degrees.

    <p>18</p> Signup and view all the answers

    Match the following real-world situations with their corresponding equations:

    <p>Seven bats and six balls cost ₹3800 = 7b + 6c = 3800 Three times a number is subtracted from another = x - 3y = 26 Taxi charges for 10 km is ₹105 = 10x + f = 105</p> Signup and view all the answers

    What do the taxi charges for 15 km amount to if for 10 km the charge is ₹105?

    <p>₹155</p> Signup and view all the answers

    If the equation 0.2x + 0.3y = 1.3 is solved for x and y, it will always yield integer solutions.

    <p>False</p> Signup and view all the answers

    What is the fixed charge for the taxi if the charge per km is assessed from the earlier data?

    <p>₹55</p> Signup and view all the answers

    What does it mean if the elimination method results in a false statement?

    <p>The equations have no solution.</p> Signup and view all the answers

    If an elimination results in a true statement involving no variable, the equations have a unique solution.

    <p>False</p> Signup and view all the answers

    What is the first step in using the elimination method?

    <p>Make the coefficients of one variable equal.</p> Signup and view all the answers

    In the elimination method, if you obtain the statement _____ involving no variable, the original pair of equations has infinitely many solutions.

    <p>true</p> Signup and view all the answers

    Match the following outcomes with their meanings in the context of equation pairs:

    <p>True statement without variables = Infinitely many solutions False statement without variables = No solution Variable equation obtained = Unique solution or continuing process Same left-hand side but different right-hand side = Inconsistent system</p> Signup and view all the answers

    When reversing the digits of a two-digit number, what happens to the ten's and unit's digits?

    <p>The unit's digit becomes the ten's digit.</p> Signup and view all the answers

    What is the general form of a two-digit number where 'x' is the ten's digit and 'y' is the unit's digit?

    <p>10x + y</p> Signup and view all the answers

    In Example 9, multiplying the first equation by 2 results in valid linear equations to work with.

    <p>True</p> Signup and view all the answers

    What is the charge for each extra day for the book Saritha and Susy borrowed?

    <p>₹3</p> Signup and view all the answers

    If two lines are parallel, the equations they represent are consistent.

    <p>False</p> Signup and view all the answers

    What method can be used to determine the unique solution of two linear equations?

    <p>Graphical method</p> Signup and view all the answers

    When the equations of two lines coincide, they have __________ solutions.

    <p>infinitely many</p> Signup and view all the answers

    Match the following situations of linear equations with their outcomes:

    <p>Two lines intersecting at a point = Unique solution Two lines coinciding = Infinitely many solutions Two parallel lines = No solution Dependent equations = Infinitely many solutions</p> Signup and view all the answers

    What is the value of x when solving the equations x + y = 6 and x - y = 2?

    <p>4</p> Signup and view all the answers

    The numbers 42 and 24 both sum to 66 and have digits that differ by 2.

    <p>True</p> Signup and view all the answers

    What are the two numbers that satisfy the equations derived from the initial problem?

    <p>42 and 24</p> Signup and view all the answers

    The sum of the digits of a two-digit number is _____ when the number is 9.

    <p>9</p> Signup and view all the answers

    Match the following methods with their corresponding equations:

    <p>x + y = 5 = Substitution Method 2x - 3y = 4 = Elimination Method 3x + 4y = 10 = Elimination Method 2x - 2y = 2 = Substitution Method</p> Signup and view all the answers

    If a fraction reduces to 1 by adding 1 to the numerator and subtracting 1 from the denominator, what is the reduced equation?

    <p>x + 1 / y - 1 = 1</p> Signup and view all the answers

    Five years ago, Nuri was twice as old as Sonu.

    <p>False</p> Signup and view all the answers

    Meena received a total of _____ notes from the cashier.

    <p>25</p> Signup and view all the answers

    Study Notes

    Pair of Linear Equations in Two Variables

    • A pair of linear equations in two variables can be represented by two lines.
    • If the lines intersect at a point, the point represents the unique solution of the two equations.
    • This is called a consistent pair of linear equations.
    • If the lines coincide, there are infinitely many solutions. Each point on the line is a solution. This is called a dependent (consistent) pair of linear equations.
    • If the lines are parallel, the pair of equations has no solution. This is called an inconsistent pair of linear equations.

    Solving Pair of Linear Equations

    • Graphical Method:
      • Requires plotting the lines represented by the equations.
      • The point of intersection is the solution.
    • Substitution Method:
      • Solve one equation for one variable.
      • Substitute the expression for the variable into the other equation.
      • Solve the resulting equation for the remaining variable.
      • Substitute the found value back into one of the original equations to find the other variable.
    • Elimination Method:
      • Multiply the equations by constants so that the coefficients of one variable are the same or opposite.
      • Add or subtract the equations to eliminate one variable.
      • Solve the resulting equation in one variable.
      • Substitute the value of that variable into one of the original equations to find the other variable.

    Special Cases:

             a₁b₁
    • Case 1:
      • ≠ a₂b₁
      • This indicates a consistent pair of linear equations. a₁b₁c₁
    • Case 2:
      • = a₂b₂ ≠ c₂
      • This indicates an inconsistent pair of linear equations. a₁b₁c₁
    • Case 3:
      • = a₂b₂ = c₂
      • This indicates a dependent and consistent pair of linear equations.

    Real-World Applications of Linear Equations

    • Linear equations can model various real-world problems like determining the number of rides and games at a fair, finding the age of individuals, or calculating the cost of items.

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    Pair Of Linear Equations In Two Variables PDF

    Description

    Test your understanding of pairs of linear equations in two variables through various solving methods. This quiz covers graphical and substitution methods, along with the concepts of consistent, dependent, and inconsistent equations. Prepare to enhance your knowledge and problem-solving skills!

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