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# Solve the following system of equations. x + 2y − 6 = z; 3y − 2z = 7; 4 + 3x = 2y − 5z.

The given system of equations are linear equations in three variables. The standard form of a three variable linear equation is ax + by + cz = k where a,b,c are coefficients, x, y, z are variables and k is a constant.

## Answer: For the following system of equations x + 2y − 6 = z; 3y − 2z = 7; 4 + 3x = 2y − 5z, the values of x, y and z are 7/ 4, 3/ 2 and - 5/ 4 respectively.

Let's solve it step by step.

**Explanation:**

Given x + 2y − 6 = z; 3y − 2z = 7; 4 + 3x = 2y − 5z.

Let, x + 2y − z = 6 ----------------- (1)

3y − 2z = 7 ----------------- (2)

3x - 2y + 5z = - 4 ----------------- (3)

Multiplying (1) with 3,

(x + 2y − z = 6) × 3,

⇒ 3x + 6y - 3z = 18 ------------------ (4)

Subtracting (3) from (4),

⇒ (3x + 6y - 3z = 18) - (3x - 2y + 5z = - 4)

⇒ (3x - 3x) + (6y + 2y) + (-3z - 5z) = 18 + 4

⇒ 8y - 8z = 22 --------------- (5)

Now, multiplying (2) with 4,

⇒ (3y − 2z = 7 ) × 4

⇒ 12y - 8z = 28 --------------- (6)

Subtracting (5) from (6),

⇒ (12y - 8z = 28) - (8y - 8z = 22)

⇒ (12y - 8y) + (- 8z + 8z) = 28 - 22

⇒ 4y = 6

⇒ y = 3/2

By putting y = 3/ 2 in (2) we get,

⇒3 (3/2) - 2z = 7

On Solving both sides, we get

⇒ 9 - 4z = 14

⇒ 4z = - 14 + 9

⇒ z = - 5/4

By putting the values of y = 3/2 and z = - 5/4 in (1) we get,

⇒ x + 2 (3/2) - ( - 5/4) = 6

⇒ x + 3 + 5/ 4 = 6

By taking LCM and solving this, we get

⇒ 4x + 12 + 5 = 24

⇒ 4x = 24 -17

⇒ x = 7/4

### Thus, the values of x = 7/4, y = 3/2 and z = - 5/4.

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