Question

How do you write a system of equations?

Answer 1

Hint:For writing the equation for a given data you have to assume some variable required according to the conditions given in the question, and when once you convert the given statement into the mathematical term then after that the matter is only to solve the formed equation.

Complete step by step solution:
For example lets us form the equation for a given statement which explains the relation as: “the product of “8” and the sum of a number “x” and “3””

For the given statement we can get that the question is asking for a variable say “y” such that
\[
8 = x \times y \\
y = x + 3 \\
\]
Now rearranging the above first equation \[8 = x \times y\] we get;

\[\dfrac{8}{x} = y\]

Now putting value of “y” in equation \[y = x + 3\] we get;

\[
\dfrac{8}{x} = x + 3 \\
x - \dfrac{8}{x} = - 3 \\
\dfrac{{{x^2} - 8}}{x} = 3 \\
{x^2} - 8 = 3x \\
{x^2} - 3x = 8 \\
\]
Now solving above quadratic equation we get;
\[
comparing\,with\varepsilon \,general\,quadratic\,equation\,a{x^2} + bx + c\,\,we\,get \\ a = 1,\,b = - 3,\,c = - 8 \\
u\sin g\,ShreeDharacharya\,rule\, \\

D(discriminant) = \sqrt {{b^2} - 4ac} = \sqrt {{{( - 3)}^2} - 4 \times 1 \times ( - 8)} \\

= \sqrt {9 + 32} \\

= \sqrt {41} \\
\]
\[

now,\,roots\,of\,equation\,are: \\
= \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\
= \dfrac{{ - ( - 3) + \sqrt {41} }}{{2(1)}},\,\dfrac{{ - ( - 3) - \sqrt {41} }}{{2(1)}}\,(D = \sqrt {{b^2} - 4ac} )
\\

= \dfrac{{3 + \sqrt {41} }}{2},\,\dfrac{{3 - \sqrt {41} }}{2} \\

= \dfrac{{3 + 6.40}}{2},\,\dfrac{{3 - 6.40}}{2}\,(\sqrt {41} = 6.40) \\

= \dfrac{{9.40}}{2},\,\dfrac{{ - 3.40}}{2} \\

= 4.7, - 1.7 \\
\]
Now we see that in the given question the product is non- negative hence the negative hence the negative term obtained as a root of the quadratic equation is not taken and only the positive term is accepted.

So or required value for “x” is \[4.7\]

Now value of “y” equals to:
\[
8 = x \times y \\
y = \dfrac{8}{x} \\
y = \dfrac{8}{{4.7}},\,(x = 4.7) \\
y = 1.70 \\
\]

Hence the required number is \[1.70\]

Note: The conversion of statement to the algebraic equation needs only the understanding of the statement, you should be careful about the words written over there and accordingly you can re-write it in the terms of maths. For example “of” is used for product, “ratio” is used for division, “sum” is used for addition and “difference” is used for subtraction.