Linear Equations in Two Variables
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Questions and Answers

Akhila went to a fair in her village and wanted to enjoy rides on the Giant Wheel and play Hoopla. What is Hoopla?

A game in which you throw a ring on the items kept in a stall, and if the ring covers any object completely, you get it.

What did Akhila spend in total at the fair?

₹20

If each ride cost ₹3, how many rides did Akhila take?

Let x represent the number of rides Akhila took. We can set up an equation: 3x + 4(x/2) = 20. Solve for x: 3x + 2x = 20 5x = 20 Therefore, Akhila took 4 rides. We know that the number of times she played Hoopla is half the number of rides. Akhila played Hoopla 4/2 = 2 times.

What is the cost of one game of Hoopla?

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

Why is it important to represent this situation as linear equations in two variables?

<p>Linear equations can be used to model real-world situations and help solve problems like this one, where two unknown quantities are related. Solving the equations allows us to find the values of each unknown quantity (in this case, the number of rides and Hoopla games).</p> Signup and view all the answers

A pair of linear equations which has no solution is called what?

<p>An inconsistent pair of linear equations</p> Signup and view all the answers

A pair of linear equations which are equivalent has infinitely many distinct common solutions. What is this pair of linear equations called?

<p>A dependent pair of linear equations</p> Signup and view all the answers

What are the three potential outcomes for the behavior of lines representing a pair of linear equations in two variables?

<p>The lines may intersect in a single point.</p> Signup and view all the answers

What situation does the pair of equations x - 2y = 0 and 3x + 4y - 20 = 0 represent?

<p>The lines intersect.</p> Signup and view all the answers

What situation does the pair of equations 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0 represent?

<p>The lines coincide.</p> Signup and view all the answers

What situation does the pair of equations x + 2y - 4 = 0 and 2x + 4y - 12 = 0 represent?

<p>The lines are parallel.</p> Signup and view all the answers

What is the general form of a linear equation?

<p>ax + by + c = 0</p> Signup and view all the answers

What are the coefficients of the equation ax + by + c = 0?

<p>a, b, and c</p> Signup and view all the answers

If the lines represented by the equations ax + by + c₁ = 0 and a₂x + b₂y + c₂ = 0 are intersecting, then a₁/a₂ ≠ b₁/b₂.

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

If the lines represented by the equations ax + by + c₁ = 0 and a₂x + b₂y + c₂ = 0 are coincident, then a₁/a₂ = b₁/b₂ = c₁/c₂.

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

If the lines represented by the equations ax + by + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel, then a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

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

What is the graphical method of solving a pair of linear equations?

<p>The graphical method involves plotting the lines that represent each equation on a graph and finding the point where the lines intersect. This point of intersection represents the solution to the pair of equations.</p> Signup and view all the answers

What are the algebraic methods of solving a pair of linear equations?

<p>The algebraic methods involve manipulating the equations algebraically to solve for the variables. The most common methods are the substitution method and the elimination method.</p> Signup and view all the answers

What is the substitution method of solving a pair of linear equations?

<p>The substitution method involves solving one equation for one variable in terms of the other variable. This value is then substituted into the other equation, resulting in a single-variable equation that can be solved. The solution for this variable is then substituted back into the original equation to find the value of the other variable.</p> Signup and view all the answers

What is the elimination method of solving a pair of linear equations?

<p>The elimination method involves manipulating the equations so that the coefficients of one variable match in both equations. The equations are then added or subtracted, eliminating one variable and leaving a single-variable equation that can be solved. The solution is then substituted back into one of the original equations to find the value of the other variable.</p> Signup and view all the answers

The elimination method can be used to solve any pair of linear equations.

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

The substitution method is a more convenient and efficient way to solve systems of linear equations than graphical methods and elimination methods.

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

The elimination method can be used in conjunction with the substitution method.

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

In the given scenario, Akhila should have taken more rides on the Giant Wheel to maximize her enjoyment since each ride was only ₹3 while Hoopla was much more expensive.

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

A pair of linear equations is inconsistent if the lines represented by the equations are parallel.

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

The ratios a₁, b₁, and c₁ from the equations ax + by + c₁ = 0 and a₂x + b₂y + c₂ = 0 can be used to determine if the lines represented by the equations are intersecting, parallel, or coincident.

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

Study Notes

Pair of Linear Equations in Two Variables

  • Introduction
    • Akhila attends a fair and enjoys rides and plays Hoopla. The number of Hoopla games is half the number of rides. She spends ₹20. The cost of a ride is ₹3 and a game of Hoopla costs ₹4. Determine the number of rides and games.
    • This problem introduces the need for linear equations in two variables to solve real-world situations.

Graphical Method

  • Solution of a Pair of Linear Equations

    • An inconsistent pair of linear equations has no solution.
    • A consistent pair of linear equations has at least one solution.
    • A dependent pair of linear equations has infinitely many solutions.

  • Summary of Solutions and the Behavior of Lines:

    • Lines intersect: one solution (consistent)
    • Lines are parallel: no solution (inconsistent)
    • Lines are coincident: infinitely many solutions (dependent, consistent)

  • Example Pairs:

    • x – 2y = 0 and 3x + 4y – 20 = 0 (intersecting)
    • 2x + 3y -9 =0 and 4x+6y-18 = 0 (coincident)
    • x + 2y – 4 = 0 and 2x + 4y – 12 = 0 (parallel)

  • Graphical Interpretation:

    • Using the ratios of the coefficients a₁, b₁, c₁ and a₂, b₂, c₂ in the general form ax + by + c = 0 of two equations.
    • If the ratio a₁/a₂ is not equal to b₁/b₂ the lines intersect
    • If the ratio a₁/a₂ and b₁/b₂ are equal but, not equal to c₁/c₂ the lines are parallel
    • If the ratio a₁/a₂, b₁/b₂, and c₁/c₂ are equal the lines are coincident

Algebraic Methods

  • Substitution Method
    • Find the value of one variable in terms of the other.
    • Substitute that value in the other equation.
    • Solve the resulting single-variable equation.
    • Substitute the obtained value back to find the value of the other variable
  • Elimination Method
    • Make the coefficients of one variable the same in the two equations (using multiplication)
    • Add or subtract the equations to eliminate the variable.
    • Solve for the variable.
    • Substitute the value back to find the value of the other variable.

Summary of Chapter

  • Pair of linear equations can be solved graphically or algebraically by substitution or elimination.
  • Geometrically, a pair of linear equations has either a unique solution (lines intersect), infinitely many solutions (lines coincide), or no solution (lines are parallel)
  • The algebraic methods (substitution and elimination methods) are used to determine the solution more precisely in cases where decimal and irrational numbers are involved.

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Pair Of Linear Equations In Two Variables PDF

Description

This quiz explores the concept of pair of linear equations in two variables, focusing on real-world problems and their graphical solutions. Learn how to determine the nature of solutions—consistent, inconsistent, or dependent—through various examples and methods for graphical intersection of lines.

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