Solve {l}{6I_{1}+5I_{2}=250}{5I_{1}+8I_{2}=70}

$$\left. \begin{array} { l } { 6 I _ { 1 } + 5 I _ { 2 } = 250 } \\ { 5 I _ { 1 } + 8 I _ { 2 } = 70 } \end{array} \right.$$

Solve for I_1, I_2

$$I_{1} = \frac{1650}{23} = 71\frac{17}{23} \approx 71.739130435$$

$$I_{2} = -\frac{830}{23} = -36\frac{2}{23} \approx -36.086956522$$

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.

$$6I_{1}+5I_{2}=250,5I_{1}+8I_{2}=70$$

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

$$6I_{1}+5I_{2}=250$$

Subtract 5I_{2} from both sides of the equation.

$$6I_{1}=-5I_{2}+250$$

Divide both sides by 6.

$$I_{1}=\frac{1}{6}\left(-5I_{2}+250\right)$$

Multiply \frac{1}{6} times -5I_{2}+250.

$$I_{1}=-\frac{5}{6}I_{2}+\frac{125}{3}$$

Substitute -\frac{5I_{2}}{6}+\frac{125}{3} for I_{1} in the other equation, 5I_{1}+8I_{2}=70.

$$5\left(-\frac{5}{6}I_{2}+\frac{125}{3}\right)+8I_{2}=70$$

Multiply 5 times -\frac{5I_{2}}{6}+\frac{125}{3}.

$$-\frac{25}{6}I_{2}+\frac{625}{3}+8I_{2}=70$$

Add -\frac{25I_{2}}{6} to 8I_{2}.

$$\frac{23}{6}I_{2}+\frac{625}{3}=70$$

Subtract \frac{625}{3} from both sides of the equation.

$$\frac{23}{6}I_{2}=-\frac{415}{3}$$

Divide both sides of the equation by \frac{23}{6}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$I_{2}=-\frac{830}{23}$$

Substitute -\frac{830}{23} for I_{2} in I_{1}=-\frac{5}{6}I_{2}+\frac{125}{3}. Because the resulting equation contains only one variable, you can solve for I_{1} directly.

$$I_{1}=-\frac{5}{6}\left(-\frac{830}{23}\right)+\frac{125}{3}$$

Multiply -\frac{5}{6} times -\frac{830}{23} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$I_{1}=\frac{2075}{69}+\frac{125}{3}$$

Add \frac{125}{3} to \frac{2075}{69} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$I_{1}=\frac{1650}{23}$$

The system is now solved.

$$I_{1}=\frac{1650}{23},I_{2}=-\frac{830}{23}$$

Steps Using Matrices

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

$$6I_{1}+5I_{2}=250,5I_{1}+8I_{2}=70$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&5\\5&8\end{matrix}\right)\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=\left(\begin{matrix}250\\70\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&5\\5&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&5\\5&8\end{matrix}\right))\left(\begin{matrix}6&5\\5&8\end{matrix}\right)\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\5&8\end{matrix}\right))\left(\begin{matrix}250\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\5&8\end{matrix}\right))\left(\begin{matrix}250\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}6&5\\5&8\end{matrix}\right))\left(\begin{matrix}250\\70\end{matrix}\right)$$

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.

$$\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{8}{6\times 8-5\times 5}&-\frac{5}{6\times 8-5\times 5}\\-\frac{5}{6\times 8-5\times 5}&\frac{6}{6\times 8-5\times 5}\end{matrix}\right)\left(\begin{matrix}250\\70\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{8}{23}&-\frac{5}{23}\\-\frac{5}{23}&\frac{6}{23}\end{matrix}\right)\left(\begin{matrix}250\\70\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{8}{23}\times 250-\frac{5}{23}\times 70\\-\frac{5}{23}\times 250+\frac{6}{23}\times 70\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}I_{1}\\I_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1650}{23}\\-\frac{830}{23}\end{matrix}\right)$$

Extract the matrix elements I_{1} and I_{2}.

$$I_{1}=\frac{1650}{23},I_{2}=-\frac{830}{23}$$

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.

$$6I_{1}+5I_{2}=250,5I_{1}+8I_{2}=70$$

To make 6I_{1} and 5I_{1} equal, multiply all terms on each side of the first equation by 5 and all terms on each side of the second by 6.

$$5\times 6I_{1}+5\times 5I_{2}=5\times 250,6\times 5I_{1}+6\times 8I_{2}=6\times 70$$

Simplify.

$$30I_{1}+25I_{2}=1250,30I_{1}+48I_{2}=420$$

Subtract 30I_{1}+48I_{2}=420 from 30I_{1}+25I_{2}=1250 by subtracting like terms on each side of the equal sign.

$$30I_{1}-30I_{1}+25I_{2}-48I_{2}=1250-420$$

Add 30I_{1} to -30I_{1}. Terms 30I_{1} and -30I_{1} cancel out, leaving an equation with only one variable that can be solved.

$$25I_{2}-48I_{2}=1250-420$$

Add 25I_{2} to -48I_{2}.

$$-23I_{2}=1250-420$$

Add 1250 to -420.

$$-23I_{2}=830$$

Divide both sides by -23.

$$I_{2}=-\frac{830}{23}$$

Substitute -\frac{830}{23} for I_{2} in 5I_{1}+8I_{2}=70. Because the resulting equation contains only one variable, you can solve for I_{1} directly.

$$5I_{1}+8\left(-\frac{830}{23}\right)=70$$

Multiply 8 times -\frac{830}{23}.

$$5I_{1}-\frac{6640}{23}=70$$

Add \frac{6640}{23} to both sides of the equation.

$$5I_{1}=\frac{8250}{23}$$

Divide both sides by 5.

$$I_{1}=\frac{1650}{23}$$

The system is now solved.

$$I_{1}=\frac{1650}{23},I_{2}=-\frac{830}{23}$$