Question

Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." Represent this situation algebraically and graphically.


Represent = Don't Represent​

Answer 1

${\large{\underline{\pmb{\sf{Given...}}}}}$

★ Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then

★ Three years from now, I shall be three times as old as you will be

${\large{\underline{\pmb{\sf{To \; Find...}}}}}$

★ Represent this situation algebraically and graphically

${\large{\underline{\pmb{\sf{Understanding \; the \; concept...}}}}}$

☀️ Concept : here we have been provided with two statements related to the ages of aftab and his daughter which are that,

⠀⠀⠀» Seven years ago, Aftab was seven times as old as his child was then

⠀⠀⠀» Three years from now, Aftab shall be three times as old as his daughter will be

❍ Now let's frame equations according to the statement assigning suitable variables to the age of Aftab and his daughter as they are undefined and then use substitution  method to solve them.

${\large{\underline{\pmb{\sf{Full\; Solution...}}}}}$

✰ Let us assume that,

⠀⠀⠀⠀⠀⋆ The present age of Aftab be x

⠀⠀⠀⠀⠀⋆  The present age of Aftab's daughter be y

~ Now let's frame an equation according to the first condition

$\longrightarrow \tt ( x - 7 ) = 7(y - 7 )$

~ Now let's simplify the equation further

$\longrightarrow \tt x - 7 = 7y - 49$

$\longrightarrow \tt x = 7y - 49 - 7$

$\longrightarrow \tt x = 7y - 42 ---(1)$

~ Now let's frame an equation according to condition 2

$\longrightarrow \tt ( x + 3 ) = 3( y + 3 )$

~ Now let's simplify the equation in further

$\longrightarrow \tt ( x + 3 ) = 3( y + 3 )$

$\longrightarrow \tt x + 3 = 3y + 9$

$\longrightarrow \tt x = 3y + 9 - 3$

$\longrightarrow \tt x - 3y = 6 ---(2)$

• Henceforth the equations are x = 7y - 42 and x - 3y = 6

~ Now let's represent the equations graphically

${\bigstar{\underline{\purple{\underline{\sf{ Table \; for \; equation 1 }}}}}}$

⋆ x - 7y = - 42

$\: \: \begin{gathered}\footnotesize\begin{gathered}\boxed{\begin{array}{ c |c |c|c } \sf \pmb{\sf x}& \sf {\pmb{ \textsf{ -} \;42}} &\sf {\pmb{ 0}} &\sf {\pmb{ \textsf{42} }}\\ \dfrac{\qquad\qquad}{ \sf {\pmb{ \sf y}}}&\dfrac{\qquad\qquad}{ \sf {\pmb{\textsf { 0}}}}&\dfrac{\qquad\qquad}{ \sf{ \pmb{6}}}& \dfrac{\qquad\qquad}{ \sf{ \pmb{ \textsf {1 }2}}}&\\ \end{array}}\end{gathered}\end{gathered}$

${\bigstar{\underline{\purple{\underline{\sf{ Table \; for \; equation \;2 }}}}}}$

⋆ x - 3y = 6

$\: \: \begin{gathered}\footnotesize\begin{gathered}\boxed{\begin{array}{ c |c |c|c } \sf \pmb{\sf x}& \sf {\pmb{ \textsf{ }0}} &\sf {\pmb{ 6}} &\sf {\pmb{ \textsf{42} }}\\ \dfrac{\qquad\qquad}{ \sf {\pmb{ \sf y}}}&\dfrac{\qquad\qquad}{ \sf {\pmb{\textsf { - }2}}}&\dfrac{\qquad\qquad}{ \sf{ \pmb{0}}}& \dfrac{\qquad\qquad}{ \sf{ \pmb{ \textsf {1 }2}}}&\\ \end{array}}\end{gathered}\end{gathered}$

→ Steps to represent the equation graphically

⋆ Take any random value and substitute it the the x cordination in the equation

⋆ Later find the value of the other variable y and mark the points on the graph

* Note : The cordinates of the 1st equation are given in the first attachment and of the second one in the 2nd as I was unable to locate all of them at once

Answer 2

Answer:

Let

age of Aftab is represented by x

age of daughter is represented by y

Then Seven years ago,

age of Aftab = x -7

age of daughter = y-7

given that I was seven times as old as you were so we get

x - 7 = 7 ( y – 7 )

x – 7 = 7 y – 49

x- 7 y = - 49 + 7

x – 7 y = - 42 …(1)

x = 7 y – 42

plug y = 5,6 and 7 we get

x = 7 * 5 - 42 = 35 - 42 = - 7

x = 7 * 6 - 42 = 42 – 42 =0

x = 7 * 7 – 42 = 49 – 42 = 7

three years from now ,

Age of Aftab = x +3

Age of daughter = y+3

Given that I shall be three times as old as you will be so we get

x + 3 = 3 (y + 3)

x + 3 = 3 y + 9

x -3 y = 9- 3

x -3 y = 6 …(2)

x = 3 y + 6

plug y = -2,-1and 0 we get

x = 3 * - 2 + 6 = -6+6 =0

x = 3 * - 1 + 6 = -3+ 6 = 3

x = 3 * 0 + 6 = 0 + 6 = 6

Algebraic representation

From equation (1) and (2)

x – 7 y = – 42 …(1)

x - 3 y = 6 …(2)

Graphical representation....on top