Question

indice #2 :
4 fois la valeur du 2e nombre augmenté de 2 fois la valeur du 3e nombre donne 278 mais le 2e nombre augmenté du 3e nombre donne 102.

indice #3 :
la valeur du 4e nombre correspond à la valeur initiale de léquation de cette droite.
(avec un graphique de droite passant par (12, 90) et (28, 202))

indice #4 :
le 5e nombre est la valeur du w dans léquation suivante :
$(4w - 3)(3w + 7) + (3w(-4w + 2)) = 979$

indice #5 :
quatre amis, al,bo, mo et to ont chacun un nombre dans leur tête. on sait que le nombre de al vaut 56 de moins que le nombre de to. celui de to vaut le triple de celui de mo. et celui de bo vaut 12 de plus que celui de mo. on sait également que la somme des nombres de al et bo est équivalente à la différence de to et mo.
le 6e nombre est celui de bo.

Explanation:

2: Simplify the equation
\( 4x + 204 - 2x = 278 \)
\( 2x + 204 = 278 \)
\( 2x = 278 - 204 \)
\( 2x = 74 \)
\( x = 37 \) (so the 2nd number is 37)

Then, \( y = 102 - 37 = 65 \) (3rd number is 65)

Indice #3:

We need to find the equation of the line passing through \( (12, 90) \) and \( (28, 202) \).

First, calculate the slope \( m \):

Step 1: Calculate the slope

\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{202 - 90}{28 - 12} = \frac{112}{16} = 7 \)

Now, use the point-slope form \( y - y_1 = m(x - x_1) \) with \( (12, 90) \):

Step 2: Find the equation of the line

\( y - 90 = 7(x - 12) \)
\( y - 90 = 7x - 84 \)
\( y = 7x + 6 \)

The "initial value" (y-intercept) is 6. So the 4th number is 6.

Indice #4:

Solve the equation \( (4w - 3)(3w + 7) + (3w(-4w + 2)) = 979 \)

Step 1: Expand the terms

First, expand \( (4w - 3)(3w + 7) \):
\( 4w \cdot 3w + 4w \cdot 7 - 3 \cdot 3w - 3 \cdot 7 = 12w^2 + 28w - 9w - 21 = 12w^2 + 19w - 21 \)

Then, expand \( 3w(-4w + 2) \):
\( -12w^2 + 6w \)

Now, add the two expanded expressions:

Step 2: Combine like terms

\( (12w^2 + 19w - 21) + (-12w^2 + 6w) = 25w - 21 \)

Set equal to 979:

Step 3: Solve for \( w \)

\( 25w - 21 = 979 \)
\( 25w = 979 + 21 \)
\( 25w = 1000 \)
\( w = 40 \)

So the 5th number is 40.

Indice #5:

Let \( \text{Mo} = m \), \( \text{To} = 3m \) (since To is triple Mo), \( \text{Al} = 3m - 56 \) (Al is 56 less than To), \( \text{Bo} = m + 12 \) (Bo is 12 more than Mo).

We also know that \( \text{Al} + \text{Bo} = \text{To} - \text{Mo} \)

Step 1: Substitute the expressions into the equation

\( (3m - 56) + (m + 12) = 3m - m \)

Step 2: Simplify and solve for \( m \)

\( 4m - 44 = 2m \)
\( 4m - 2m = 44 \)
\( 2m = 44 \)
\( m = 22 \)

Now, find Bo: \( \text{Bo} = 22 + 12 = 34 \)

So the 6th number (Bo) is 34.

Summary of Numbers:

• 2nd number: 37
• 3rd number: 65
• 4th number: 6
• 5th number: 40
• 6th number: 34

Wait, but we need to confirm if we missed the 1st number? Wait, the indices start at #2, so maybe the 1st number is not needed yet. Let's check if we need to find more, but the problem might be to find all numbers. Wait, maybe the original problem is to find a code or a sequence, but since the user provided the indices, let's list the numbers we found:

• 2nd: 37
• 3rd: 65
• 4th: 6
• 5th: 40
• 6th: 34

If we need to present all, but maybe the question was to solve each index. Let's confirm the solutions:

• Indice #2: 2nd number = 37, 3rd = 65
• Indice #3: 4th number = 6
• Indice #4: 5th number = 40
• Indice #5: 6th number = 34

If you need the 1st number, maybe it's not given, but based on the indices, we solved the ones provided.

Answer:

2: Simplify the equation
\( 4x + 204 - 2x = 278 \)
\( 2x + 204 = 278 \)
\( 2x = 278 - 204 \)
\( 2x = 74 \)
\( x = 37 \) (so the 2nd number is 37)

Then, \( y = 102 - 37 = 65 \) (3rd number is 65)

Indice #3:

We need to find the equation of the line passing through \( (12, 90) \) and \( (28, 202) \).

First, calculate the slope \( m \):

Step 1: Calculate the slope

\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{202 - 90}{28 - 12} = \frac{112}{16} = 7 \)

Now, use the point-slope form \( y - y_1 = m(x - x_1) \) with \( (12, 90) \):

Step 2: Find the equation of the line

\( y - 90 = 7(x - 12) \)
\( y - 90 = 7x - 84 \)
\( y = 7x + 6 \)

The "initial value" (y-intercept) is 6. So the 4th number is 6.

Indice #4:

Solve the equation \( (4w - 3)(3w + 7) + (3w(-4w + 2)) = 979 \)

Step 1: Expand the terms

First, expand \( (4w - 3)(3w + 7) \):
\( 4w \cdot 3w + 4w \cdot 7 - 3 \cdot 3w - 3 \cdot 7 = 12w^2 + 28w - 9w - 21 = 12w^2 + 19w - 21 \)

Then, expand \( 3w(-4w + 2) \):
\( -12w^2 + 6w \)

Now, add the two expanded expressions:

Step 2: Combine like terms

\( (12w^2 + 19w - 21) + (-12w^2 + 6w) = 25w - 21 \)

Set equal to 979:

Step 3: Solve for \( w \)

\( 25w - 21 = 979 \)
\( 25w = 979 + 21 \)
\( 25w = 1000 \)
\( w = 40 \)

So the 5th number is 40.

Indice #5:

Let \( \text{Mo} = m \), \( \text{To} = 3m \) (since To is triple Mo), \( \text{Al} = 3m - 56 \) (Al is 56 less than To), \( \text{Bo} = m + 12 \) (Bo is 12 more than Mo).

We also know that \( \text{Al} + \text{Bo} = \text{To} - \text{Mo} \)

Step 1: Substitute the expressions into the equation

\( (3m - 56) + (m + 12) = 3m - m \)

Step 2: Simplify and solve for \( m \)

\( 4m - 44 = 2m \)
\( 4m - 2m = 44 \)
\( 2m = 44 \)
\( m = 22 \)

Now, find Bo: \( \text{Bo} = 22 + 12 = 34 \)

So the 6th number (Bo) is 34.

Summary of Numbers:

• 2nd number: 37
• 3rd number: 65
• 4th number: 6
• 5th number: 40
• 6th number: 34

Wait, but we need to confirm if we missed the 1st number? Wait, the indices start at #2, so maybe the 1st number is not needed yet. Let's check if we need to find more, but the problem might be to find all numbers. Wait, maybe the original problem is to find a code or a sequence, but since the user provided the indices, let's list the numbers we found:

• 2nd: 37
• 3rd: 65
• 4th: 6
• 5th: 40
• 6th: 34

If we need to present all, but maybe the question was to solve each index. Let's confirm the solutions:

• Indice #2: 2nd number = 37, 3rd = 65
• Indice #3: 4th number = 6
• Indice #4: 5th number = 40
• Indice #5: 6th number = 34

If you need the 1st number, maybe it's not given, but based on the indices, we solved the ones provided.