Ken plays a game with the following constraints:

He can score $$3$$ or $$6$$ points each turn in the game.

He can take up to $$10$$ turns in the game.

He wins the game if he scores $$39$$ points or more.

Winning the game is represented by the given system of inequalities.

$$6 x+3 y \geq 39 $$
$$x+y \leq 10$$
Ken wins the game.

What is the least number of turns during which Ken must have scored $$6$$ points?




<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>Ken plays a game with the following constraints:

He can score $$3$$ or $$6$$ points each turn in the game.

He can take up to $$10$$ turns in the game.

He wins the game if he scores $$39$$ points or more.

Winning the game is represented by the given system of inequalities.

$$6 x+3 y \geq 39 $$
$$x+y \leq 10$$
Ken wins the game.

What is the least number of turns during which Ken must have scored $$6$$ points?


</textarea>Ken plays a game with the following constraints:
He can score <math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>3 or <math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math>6 points each turn in the game.
He can take up to <math><semantics><mrow><mn>1</mn><mn>0</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math>10 turns in the game.
He wins the game if he scores <math><semantics><mrow><mn>3</mn><mn>9</mn></mrow><annotation encoding="application/x-tex">39</annotation></semantics></math>39 points or more.
Winning the game is represented by the given system of inequalities.
<math><semantics><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≥</mo><mn>3</mn><mn>9</mn></mrow><annotation encoding="application/x-tex">6 x+3 y \geq 39 </annotation></semantics></math>6x+3y≥39 <math><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mn>1</mn><mn>0</mn></mrow><annotation encoding="application/x-tex">x+y \leq 10</annotation></semantics></math>x+y≤10 Ken wins the game.
What is the least number of turns during which Ken must have scored <math><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math>6 points?
</div>

<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>$$x$$ and $$y$$ be the number of turns ken playes to score 6 and 3 points respectively.

Given that, 

![](http://admin.getprephub.com/media/editor/sp20_20230223191421386736.png)

$$6 x+3 y \geq 39$$ 

or $$2 x+y \geq 13 \ldots \ldots$$ (i)

$$x+y \leq 10 \ldots \ldots \ldots$$ (ii)

Converting inequality to equality,

$$2 x+y=13 \ldots \ldots \ldots . .(i)$$ 

$$x+y=10 \ldots \ldots \ldots . . .(i i)$$ 

by $$(i)-(ii)$$,

$$2 x-x=13-10$$ 

Or, $$x=3$$

Putting $$x=3$$ in $$(ii)$$, $$y=7$$

The least number of turns during which Ken must have scored 6 points is $$3$$ .
</textarea><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>x and <math><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>y be the number of turns ken playes to score 6 and 3 points respectively.
Given that, 
<img src="http://admin.getprephub.com/media/editor/sp20_20230223191421386736.png" alt="">
<math><semantics><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≥</mo><mn>3</mn><mn>9</mn></mrow><annotation encoding="application/x-tex">6 x+3 y \geq 39</annotation></semantics></math>6x+3y≥39 
or <math><semantics><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mn>1</mn><mn>3</mn><mo separator="true">…</mo><mo separator="true">…</mo></mrow><annotation encoding="application/x-tex">2 x+y \geq 13 \ldots \ldots</annotation></semantics></math>2x+y≥13…… (i)
<math><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mn>1</mn><mn>0</mn><mo separator="true">…</mo><mo separator="true">…</mo><mo separator="true">…</mo></mrow><annotation encoding="application/x-tex">x+y \leq 10 \ldots \ldots \ldots</annotation></semantics></math>x+y≤10……… (ii)
Converting inequality to equality,
<math><semantics><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>1</mn><mn>3</mn><mo separator="true">…</mo><mo separator="true">…</mo><mo separator="true">…</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>(</mo><mi>i</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">2 x+y=13 \ldots \ldots \ldots . .(i)</annotation></semantics></math>2x+y=13………..(i) 
<math><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>1</mn><mn>0</mn><mo separator="true">…</mo><mo separator="true">…</mo><mo separator="true">…</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">x+y=10 \ldots \ldots \ldots . . .(i i)</annotation></semantics></math>x+y=10………...(ii) 
by <math><semantics><mrow><mo>(</mo><mi>i</mi><mo>)</mo><mo>−</mo><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(i)-(ii)</annotation></semantics></math>(i)−(ii),
<math><semantics><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>x</mi><mo>=</mo><mn>1</mn><mn>3</mn><mo>−</mo><mn>1</mn><mn>0</mn></mrow><annotation encoding="application/x-tex">2 x-x=13-10</annotation></semantics></math>2x−x=13−10 
Or, <math><semantics><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">x=3</annotation></semantics></math>x=3
Putting <math><semantics><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">x=3</annotation></semantics></math>x=3 in <math><semantics><mrow><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(ii)</annotation></semantics></math>(ii), <math><semantics><mrow><mi>y</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">y=7</annotation></semantics></math>y=7
The least number of turns during which Ken must have scored 6 points is <math><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>3 .
</div>