Question

3. Form the pair of linear equations for the following problems and find their solution by substitution method. (1) The difference between two numbers is 26 and one number is three times the other. Find them. (i) The larger of two supplementary angles exceeds the smaller by i 8 degrees. Find them. (ii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball. ​

Answer 1

$1)$ Answer:

  $39$ and $13$

  Step-by-step solution:

  Let the two numbers be $x$ and $y$; Let $x>y$

  The question states that: $a)$ The difference between two numbers is $26$

                                                 ⇒ $x - y = 26$    --$(1)$

                                              $b)$ one number is three times the other.

                                                  ⇒ $x = 3y$      --$(2)$

 

  From $(2)$

  We get that $x = 3y$

  Substituting the value of $x$ in $(1)$

  ⇒ $3y-y = 26$

  ⇒ $2y = 26$

  ⇒ $y = 13$

  Substituting the value of $y$ in $(2)$

  ⇒ $x = 3y = 3(13) = 39$

  ∴ The two numbers are $39$ and $13$

$i)$  Answer:

   The two angles are $86$° and $94$°

   Step-by-step Solution:

   Let the two angles be $a$ and $b$; let $a>b$

   The question states that: $a)$ The larger of two supplementary angles

                                                  Exceeds the smaller by $8$°

                                                  ⇒ $a = b+8$      --$(1)$

                                              $b)$ The angles are supplementary  

                                                  ⇒ $a+b = 180$°        --$(2)$

   From $(1)$

   We get that $a = b +8$

   Substituting the value of $a$ in $(2)$

   ⇒ $(b+8) + b = 180$

   ⇒ $2b + 8 - 180$

   ⇒ $2b = 172$

   ⇒ $b = 86$°

   Substituting the value of $b$ in $(1)$

   ⇒ $(86) + 8 = a$

   ⇒ $a = 94$°

   ∴ The two angles are $86$° and $94$°

$ii)$  Answer:

    The cost of each bat and each ball is $Rs.500$ and $Rs. 50$ respectively

    Step-by-step Solution:

    Let the price of one bat and one ball be $x$ and $y$ respectively

    The question states that: $a)$ The coach of a cricket team buys $7$ bats and

                                                   $6$ balls for $Rs.3800$

                                                   ⇒ $7x + 6y = 3800$     --$(1)$

                                                $b)$ She buys $3$ bats and $5$ balls for $Rs.1750$

                                                    ⇒ $3x + 5y = 1750$     --$(2)$

    In $(2)$

        $3x + 5y = 1750$

    ⇒ $3x = 1750-5y$

    ⇒ $x = \frac{1750 - 5y}{3}$

    Substituting the value of $x$ in $(1)$

        $7(\frac{1750-5y}{3} ) + 6y = 3800$

    ⇒ $\frac{7(1750) - 7(5y)}{3} + 6y = 3800$

    ⇒ $\frac{12250 - 35y}{3} + 6y = 3800$

    ⇒ $\frac{12250 - 35y + 18y}{3} = 3800$

    ⇒ $12250 - 17y = 3(3800)$

    ⇒ $12250 - 17y = 11400$

    ⇒ $-17y = 11400 - 12250$

    ⇒ $-17y = -850$

    ⇒ $y = 50$

    Substituting the value of $y$ in $(2)$

    ⇒ $3x + 5(50) = 1750$

    ⇒ $3x + 250 = 1750$

    ⇒ $3x = 1500$

    ⇒ $x = 500$

    ∴  The cost of each bat and each ball is $Rs.500$ and $Rs. 50$ respectively