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Problem1: [Systems of Linear Equations] Determine which of the following equations are linear

Systems of Linear Equations

A linear equation in the $n$-variables, $x_1, x_2, ...., x_n$ to be the expression of the form $a_1x_1 + a_2x_2 + .... + a_nx_n = b$ where, $a_1, a_2, ......., a_n$ and $b$ are real constants. The variables are also called the unknowns.

Problem1: [Systems of Linear Equations] Determine which of the following equations are linear. 

1. $3x-y = -7$

2. $2x_1+3x_2-4x_3+7x_4 = -1$

3. $x=\frac{1}{3}y-z+5$

4. $x_1+2x_2+...+nx_n=1$

5. $x^2+2x+3y = 5$

6. $xy-y+z_5 = 0$

7. $x-\cos y=2$

8. $\sqrt{x}+y+\sqrt{2}=3$

Solution:

1. Linear - in the form of $a_1x_1 + a_2x_2 + .... + a_nx_n = b$

2. Linear - in the form of $a_1x_1 + a_2x_2 + .... + a_nx_n = b$

3. Linear - in the form of $a_1x_1 + a_2x_2 + .... + a_nx_n = b$

4. Linear - in the form of $a_1x_1 + a_2x_2 + .... + a_nx_n = b$

5. Not linear -  contains non linear term $x^2$

6. Not linear - contains non linear term $xy$

7. Not linear- contains non linear term $\cos y$

8. Not linear- contains non linear term $\sqrt{x}$


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