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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