Question

Find the number of solutions in positive integers of \[11x + 15y = 1031\]

Answer 1

Hint: Firstly we transform the equation in such a way that we can select proper terms and combine then by picking common among them and select its particular range and then solve as per mentioned.

Complete step by step Answer:

 \[  0 = y + 2(0) - 2 \]

\[   \Rightarrow {\text{y = 2}} \]

 Now on substituting the value of z and y in \[{\text{z = x + y - 93}}\], we get,

\[  0 = {\text{x + 2 - 93}} \]

\[   \Rightarrow {\text{x = 91}} \]

Given \[11x + 15y = 1031\],

Here , we are splitting the terms of an equation in order to simplify it as below,

\[  {{\text{11x + (11y + 4y) = 11 $\times$ 93 + 8}}} \]

\[  { \Rightarrow {\text{4y + 11(x + y - 93) = 8}}}\]

And then proceed with ,

\[  {{\text{4y + 11z = 8 where z = x + y - 93}}} \]

\[  { \Rightarrow {\text{4y + (4 $\times$ 22 + 3z) = 4 $\times$ 2}}} \]

\[  { \Rightarrow {\text{3z + 4(y + 22 - 2) = 0}}}\]

Now, here again let the new term and solve it,

\[{\text{3z + 4t = 0}}\]where \[{\text{t = y + 2z - 2}}\]

Now let \[{\text{t = 0}}\]

So we get,

\[{\text{3z = 0}}\]

\[ \Rightarrow {\text{z = 0}}\]

Now on substituting the value of z and t in \[{\text{t = y + 2z - 2}}\]

We get,

So, Solutions are,

\[{\text{x = 91 + 15k ; y = 2 - 11k}}\]

For solutions in positive integers,

\[  {\dfrac{{{\text{ - 91}}}}{{{\text{15}}}}{\text{ < k < }}\dfrac{{\text{2}}}{{{\text{11}}}}} \]

\[  {{\text{or - 6}}{\text{.06 < k < 0}}{\text{.18}}} \]

\[  { \Rightarrow {\text{k = - 6, - 5, - 4, - 3, - 2, - 1,0}}} \]

So, k has \[{\text{7}}\] possible values.

Hence, there are \[{\text{7}}\] possible solutions in positive integers of the equation \[11x + 15y = 1031\].

Note: Positive and Negative Integers

Positive integers are all the whole numbers greater than zero: \[{\text{1, 2, 3, 4, 5, }}...\]

For each positive integer, there is a negative integer, and these integers are called opposites. For example, \[ - 3\] is the opposite of \[3\], \[ - 21\] is the opposite of \[21\], and \[ - 8\] is the opposite of \[8\]. An integer is colloquially defined as a number that can be written without a fractional component. The sign of an integer is either positive (+) or negative (-), except zero, which has no sign. Two integers are opposites if they are each the same distance away from zero but on opposite sides of the number line. One will have a positive sign, the other a negative sign.