Chapter 4, Problem 1E

Solve the following systems of equations:

1. (a) 2v₂ − 4v₁ = 9 and v₁ = 5v₂ = −4;
2. (b) −v₁ + 2v₃ = 8; 2v₁ + v₂ − 5v₃ = −7; 4v₁ + 5v₂ + 8v₃ = 6.

To determine

Find the value of $v_1$ and $v_2$.

Answer to Problem 1E

The value of $v_1$ is $-2.056\text{V}$ and the value $v_2$ is $0.389\text{V}$.

Explanation of Solution

Given data:

The given equations are,

$2v_2-4v_1=9$        (1)

$v_1-5v_2=-4$        (2)

Calculation:

Rearrange the equation (1) for $v_1$.

$v_1=\left(\frac{2v_2-9}{4}\right)$

Substitute $\left(\frac{2v_2-9}{4}\right)$ for $v_1$ in equation (2).

$\begin{array}{l}\left(\frac{2v_2-9}{4}\right)-5v_2=-4\\ 18v_2=7\end{array}$

Rearrange the above equations for $v_2$.

$v_2=0.389\text{V}$

Substitute 0.389 for $v_2$ in equation (2).

$\begin{array}{l}{v}_1-5\left(0.389\right)=-4\\ v_1=-4+1.945\\ v_1=-2.056\end{array}$

Conclusion:

Thus, the value of $v_1$ is $-2.056\text{V}$ and the value $v_2$ is $0.389\text{V}$.

To determine

Find the value of $v_1$, $v_2$ and. $v_3$.

Answer to Problem 1E

The value of $v_1$ is $-8.667$, the value of $v_2$ is $8.667$ and the value of $v_3$ is $-0.333$.

Explanation of Solution

Given data:

$-v_1+2v_3=8$        (3)

$2v_1+v_2-5v_3=-7$        (4)

$4v_1+5v_2+8v_3=6$        (5)

Calculation:

Rearrange the equation (3) for $v_1$.

$v_1=\left(2v_3-8\right)$        (6)

Substitute the equation (6) in equation the equation (4).

$2\left(2v_3-8\right)+v_2-5v_3=-7$        (7)

Rearrange the equation (7) for $v_2$.

$v_2=\left(9+v_3\right)$        (8)

Substitute the equation (6) and the equation (8) in equation (5).

$4\left(2v_3-8\right)+5\left(9+v_3\right)+8v_3=6$        (9)

Rearrange the equation (9) for $v_3$.

$v_3=-0.333$

Substitute equation $-0.333$ for $v_3$ in the equation (6).

$\begin{array}{c}{v}_1=\left(2\times \left(-0.333\right)-8\right)\\ =-8.667\text{V}\end{array}$

Substitute equation $-0.333$ for $v_3$ in the equation (8).

$\begin{array}{c}{v}_2=9+\left(-0.333\right)\\ =8.667\text{V}\end{array}$

Conclusion:

Thus, the value of $v_1$ is $-8.667\text{V}$, the value of $v_2$ is $8.667\text{V}$ and the value of $v_3$ is $-0.333\text{V}$.