Solve {l}{2I_{1}-11u_{1}=45}{21I_{1}-8u_{1}=55}

$$\left. \begin{array} { l } { 2 I _ { 1 } - 11 u _ { 1 } = 45 } \\ { 21 I _ { 1 } - 8 u _ { 1 } = 55 } \end{array} \right.$$

Solve for I_1, u_1

$$I_{1} = \frac{49}{43} = 1\frac{6}{43} \approx 1.139534884$$

$$u_{1} = -\frac{167}{43} = -3\frac{38}{43} \approx -3.88372093$$

Steps Using Substitution

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

$$2I_{1}-11u_{1}=45,21I_{1}-8u_{1}=55$$

Choose one of the equations and solve it for I_{1} by isolating I_{1} on the left hand side of the equal sign.

$$2I_{1}-11u_{1}=45$$

Add 11u_{1} to both sides of the equation.

$$2I_{1}=11u_{1}+45$$

Divide both sides by 2.

$$I_{1}=\frac{1}{2}\left(11u_{1}+45\right)$$

Multiply \frac{1}{2} times 11u_{1}+45.

$$I_{1}=\frac{11}{2}u_{1}+\frac{45}{2}$$

Substitute \frac{11u_{1}+45}{2} for I_{1} in the other equation, 21I_{1}-8u_{1}=55.

$$21\left(\frac{11}{2}u_{1}+\frac{45}{2}\right)-8u_{1}=55$$

Multiply 21 times \frac{11u_{1}+45}{2}.

$$\frac{231}{2}u_{1}+\frac{945}{2}-8u_{1}=55$$

Add \frac{231u_{1}}{2} to -8u_{1}.

$$\frac{215}{2}u_{1}+\frac{945}{2}=55$$

Subtract \frac{945}{2} from both sides of the equation.

$$\frac{215}{2}u_{1}=-\frac{835}{2}$$

Divide both sides of the equation by \frac{215}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$u_{1}=-\frac{167}{43}$$

Substitute -\frac{167}{43} for u_{1} in I_{1}=\frac{11}{2}u_{1}+\frac{45}{2}. Because the resulting equation contains only one variable, you can solve for I_{1} directly.

$$I_{1}=\frac{11}{2}\left(-\frac{167}{43}\right)+\frac{45}{2}$$

Multiply \frac{11}{2} times -\frac{167}{43} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$I_{1}=-\frac{1837}{86}+\frac{45}{2}$$

Add \frac{45}{2} to -\frac{1837}{86} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$I_{1}=\frac{49}{43}$$

The system is now solved.

$$I_{1}=\frac{49}{43},u_{1}=-\frac{167}{43}$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$2I_{1}-11u_{1}=45,21I_{1}-8u_{1}=55$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right)\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=\left(\begin{matrix}45\\55\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&-11\\21&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right))\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right)\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right))\left(\begin{matrix}45\\55\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right))\left(\begin{matrix}45\\55\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=inverse(\left(\begin{matrix}2&-11\\21&-8\end{matrix}\right))\left(\begin{matrix}45\\55\end{matrix}\right)$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.

$$\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{2\left(-8\right)-\left(-11\times 21\right)}&-\frac{-11}{2\left(-8\right)-\left(-11\times 21\right)}\\-\frac{21}{2\left(-8\right)-\left(-11\times 21\right)}&\frac{2}{2\left(-8\right)-\left(-11\times 21\right)}\end{matrix}\right)\left(\begin{matrix}45\\55\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{215}&\frac{11}{215}\\-\frac{21}{215}&\frac{2}{215}\end{matrix}\right)\left(\begin{matrix}45\\55\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{215}\times 45+\frac{11}{215}\times 55\\-\frac{21}{215}\times 45+\frac{2}{215}\times 55\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}I_{1}\\u_{1}\end{matrix}\right)=\left(\begin{matrix}\frac{49}{43}\\-\frac{167}{43}\end{matrix}\right)$$

Extract the matrix elements I_{1} and u_{1}.

$$I_{1}=\frac{49}{43},u_{1}=-\frac{167}{43}$$

Steps Using Elimination

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$2I_{1}-11u_{1}=45,21I_{1}-8u_{1}=55$$

To make 2I_{1} and 21I_{1} equal, multiply all terms on each side of the first equation by 21 and all terms on each side of the second by 2.

$$21\times 2I_{1}+21\left(-11\right)u_{1}=21\times 45,2\times 21I_{1}+2\left(-8\right)u_{1}=2\times 55$$

Simplify.

$$42I_{1}-231u_{1}=945,42I_{1}-16u_{1}=110$$

Subtract 42I_{1}-16u_{1}=110 from 42I_{1}-231u_{1}=945 by subtracting like terms on each side of the equal sign.

$$42I_{1}-42I_{1}-231u_{1}+16u_{1}=945-110$$

Add 42I_{1} to -42I_{1}. Terms 42I_{1} and -42I_{1} cancel out, leaving an equation with only one variable that can be solved.

$$-231u_{1}+16u_{1}=945-110$$

Add -231u_{1} to 16u_{1}.

$$-215u_{1}=945-110$$

Add 945 to -110.

$$-215u_{1}=835$$

Divide both sides by -215.

$$u_{1}=-\frac{167}{43}$$

Substitute -\frac{167}{43} for u_{1} in 21I_{1}-8u_{1}=55. Because the resulting equation contains only one variable, you can solve for I_{1} directly.

$$21I_{1}-8\left(-\frac{167}{43}\right)=55$$

Multiply -8 times -\frac{167}{43}.

$$21I_{1}+\frac{1336}{43}=55$$

Subtract \frac{1336}{43} from both sides of the equation.

$$21I_{1}=\frac{1029}{43}$$

Divide both sides by 21.

$$I_{1}=\frac{49}{43}$$

The system is now solved.

$$I_{1}=\frac{49}{43},u_{1}=-\frac{167}{43}$$