The following is a system of linear equations
x - 2y + z = 34 (1)
2x + y + z = 102 (2)
x + y - 3z = 17 (3)
The value of \( x + y + z \) is ________. (rounded off to two decimal places)
Step 1: Solve the system of equations. We are given:
(1) x - 2y + z = 34
(2) 2x + y + z = 102
(3) x + y - 3z = 17
We will solve this using substitution or elimination.
Step 2: Eliminate one variable. Let’s eliminate \( z \) from equations (1) and (2). Subtract (1) from (2):
(2) - (1): (2x + y + z) - (x - 2y + z) = 102 - 34
x + 3y = 68 ... (4)
Now eliminate \( z \) from equations (1) and (3): Multiply (1) by 3 and add to (3):
3(x - 2y + z) = 3 * 34 = 102 => 3x - 6y + 3z = 102
(3) + 3 * (1): (x + y - 3z) + (3x - 6y + 3z) = 17 + 102
4x - 5y = 119 ... (5)
Step 3: Solve equations (4) and (5): From (4): \( x = 68 - 3y \) Substitute into (5):
4(68 - 3y) - 5y = 119
272 - 12y - 5y = 119
272 - 17y = 119 => 17y = 153 => y = 9
Now, substitute \( y = 9 \) into (4):
x + 3(9) = 68 => x = 68 - 27 = 41
Now substitute \( x = 41 \), \( y = 9 \) into (1):
41 - 2(9) + z = 34 => 41 - 18 + z = 34 => z = 11
Step 4: Compute \( x + y + z \)
x + y + z = 41 + 9 + 11 = 61
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?