Linear Equation in Three Variables

**Category : **10th Class

We can also use the system of linear equation for solving the linear equation in three variables by mere substitution. In this method we find one of the three variables in terms of other two from any one of the equation and substitute it in the second equation. From the second equation we obtained the second variable in terms of the other and substitute it in the third equation and solve it to get the third variable which on re-substituting in the previous steps we get the other variables. It can also be solved by elimination method.

** Solve the following system of linear equation: \[3x-y+4z=3,x+2y-3z=0\, and \,6x+5y=-3.\] **

(a) \[x=-\frac{39}{10},y=\frac{47}{10},z=\frac{5}{2}\]

(b) \[x=\frac{39}{10},y=\frac{47}{10},z=\frac{5}{2}\]

(c) \[x=\frac{39}{10},y=\frac{7}{10},z=\frac{25}{2}\]

(d) \[x=-\frac{9}{10},y=\frac{17}{10},z=\frac{25}{2}\]

(e) None of these

**Answer: (e)**

**Explanation**

We have,

\[3x-y+4z=3-----\left( 1 \right)\]

\[x-2y-3z=-2-----\left( 2 \right)\]

\[6x+5y-5z=-3-----\left( 3 \right)\]

Form equation (1), we have

\[-y=3-3x-4x-----\left( 4 \right)\]

Putting in equation (2) the above value we get,

\[x=\frac{8-13z}{5}-----\left( 5 \right)\]

Putting equation (5) in (4) we get,

\[y=\frac{-9+13z}{5}-----\left( 6 \right)\]

** Solve the following system of linear equation: \[5x-7y+z=11,6x-8y-z=15\,and\,3x+2y-6z=7.\]**

(a) \[x=2,y=3,z=2\]

(b) \[x=-2,y=5,z=-1\]

(c) \[x=1,y=-1,z=-1\]

(d) \[x=-3,y=1,z=-2\]

(e) None of these

**Answer: (c)**

**Solve the system of the equation: \[6x+y-3z=5,x+3y-2z=5,2x+y+4z=8\]**

(a) \[x=1,y=2,z=1\]

(b) \[x=-2,y=5,z=-1\]

(c) \[x=1,y=-1,z=-1\]

(d) \[x=-3,y=1,z=-2\]

(e) None of these

** **

**Answer: (a)**

**Solve the system of equation: \[2y-3z=0,x+3y=-4,3x+4y=3\]**

(a) \[x=2,y=3,z=2\]

(b) \[x=5,y=-3,z=-2\]

(c) \[x=4,y=-3,z=-1\]

(d) \[x=-2,y=2,z=-2\]

(e) None of these

** **

**Answer: (b)**

**Find the solution of the system of the equation: \[x+y=8,y+z=10,x+z=12\]**

(a) \[x=5,y=3,z=2\]

(b) \[x=5,y=5,z=-1\]

(c) \[x=8,y=-2,z=-1\]

(d) \[x=5,y=3,z=7\]

(e) None of these

**Answer: (d)**

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- The 6th Euler number is 61.
- 65 is the smallest number that becomes square if its reverse is either added to or subtracted from it.
- The smallest square which is the sum of 4 consecutive squares is 100.
- 8 years, 7 months & 6 days of yelling creates enough sound energy to heat 1 cup of coffee.
- Prior to 1900, prize fights lasted up to 100 rounds. Butterflies taste with their feet!

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- Two equations in same variable are called pair of linear equation.
- A pair of linear equation can be solved by different methods: Such as substitution method, elimination method, and cross multiplication method.
- It can also be solved by graphical method.
- If the pair of line intersects each other at one point, then it is said to have unique solution and the point of intersection is the solution of the pair of equation.
- If the pair of line is coincident, then the equation is said to have no solution.
- If the pair of line is coincident, then the equation is said to have infinitely many solution.
- If the equation has no solution, then it is said to be inconsistent, other- wise it is said to be consistent.
- There are various situations which can be mathematically represented by two equations that are not linear to start with.
- It can be used to solve the daily life problem reducing them in the form of linear equation.

*play_arrow*Introduction*play_arrow*Graphical Representation*play_arrow*Elimination Method of Solving*play_arrow*Condition for Consistency*play_arrow*Linear Equation in Three Variables*play_arrow*Linear Equation in two Variables

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