Solve {l}{100x_{1}+30x_{2}=30}{75x_{1}-45x_{2}=20}

$$\left. \begin{array} { l } { 100 x _ { 1 } + 30 x _ { 2 } = 30 } \\ { 75 x _ { 1 } - 45 x _ { 2 } = 20 } \end{array} \right.$$

Solve for x_1, x_2

$$x_{1}=\frac{13}{45}\approx 0.288888889$$

$$x_{2}=\frac{1}{27}\approx 0.037037037$$

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.

$$100x_{1}+30x_{2}=30,75x_{1}-45x_{2}=20$$

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

$$100x_{1}+30x_{2}=30$$

Subtract 30x_{2} from both sides of the equation.

$$100x_{1}=-30x_{2}+30$$

Divide both sides by 100.

$$x_{1}=\frac{1}{100}\left(-30x_{2}+30\right)$$

Multiply \frac{1}{100} times -30x_{2}+30.

$$x_{1}=-\frac{3}{10}x_{2}+\frac{3}{10}$$

Substitute \frac{-3x_{2}+3}{10} for x_{1} in the other equation, 75x_{1}-45x_{2}=20.

$$75\left(-\frac{3}{10}x_{2}+\frac{3}{10}\right)-45x_{2}=20$$

Multiply 75 times \frac{-3x_{2}+3}{10}.

$$-\frac{45}{2}x_{2}+\frac{45}{2}-45x_{2}=20$$

Add -\frac{45x_{2}}{2} to -45x_{2}.

$$-\frac{135}{2}x_{2}+\frac{45}{2}=20$$

Subtract \frac{45}{2} from both sides of the equation.

$$-\frac{135}{2}x_{2}=-\frac{5}{2}$$

Divide both sides of the equation by -\frac{135}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x_{2}=\frac{1}{27}$$

Substitute \frac{1}{27} for x_{2} in x_{1}=-\frac{3}{10}x_{2}+\frac{3}{10}. Because the resulting equation contains only one variable, you can solve for x_{1} directly.

$$x_{1}=-\frac{3}{10}\times \frac{1}{27}+\frac{3}{10}$$

Multiply -\frac{3}{10} times \frac{1}{27} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x_{1}=-\frac{1}{90}+\frac{3}{10}$$

Add \frac{3}{10} to -\frac{1}{90} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x_{1}=\frac{13}{45}$$

The system is now solved.

$$x_{1}=\frac{13}{45},x_{2}=\frac{1}{27}$$

Steps Using Matrices

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

$$100x_{1}+30x_{2}=30,75x_{1}-45x_{2}=20$$

Write the equations in matrix form.

$$\left(\begin{matrix}100&30\\75&-45\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}30\\20\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}100&30\\75&-45\end{matrix}\right).

$$inverse(\left(\begin{matrix}100&30\\75&-45\end{matrix}\right))\left(\begin{matrix}100&30\\75&-45\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}100&30\\75&-45\end{matrix}\right))\left(\begin{matrix}30\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}100&30\\75&-45\end{matrix}\right))\left(\begin{matrix}30\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=inverse(\left(\begin{matrix}100&30\\75&-45\end{matrix}\right))\left(\begin{matrix}30\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}-\frac{45}{100\left(-45\right)-30\times 75}&-\frac{30}{100\left(-45\right)-30\times 75}\\-\frac{75}{100\left(-45\right)-30\times 75}&\frac{100}{100\left(-45\right)-30\times 75}\end{matrix}\right)\left(\begin{matrix}30\\20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{150}&\frac{1}{225}\\\frac{1}{90}&-\frac{2}{135}\end{matrix}\right)\left(\begin{matrix}30\\20\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{1}{150}\times 30+\frac{1}{225}\times 20\\\frac{1}{90}\times 30-\frac{2}{135}\times 20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x_{1}\\x_{2}\end{matrix}\right)=\left(\begin{matrix}\frac{13}{45}\\\frac{1}{27}\end{matrix}\right)$$

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

$$x_{1}=\frac{13}{45},x_{2}=\frac{1}{27}$$

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.

$$100x_{1}+30x_{2}=30,75x_{1}-45x_{2}=20$$

To make 100x_{1} and 75x_{1} equal, multiply all terms on each side of the first equation by 75 and all terms on each side of the second by 100.

$$75\times 100x_{1}+75\times 30x_{2}=75\times 30,100\times 75x_{1}+100\left(-45\right)x_{2}=100\times 20$$

Simplify.

$$7500x_{1}+2250x_{2}=2250,7500x_{1}-4500x_{2}=2000$$

Subtract 7500x_{1}-4500x_{2}=2000 from 7500x_{1}+2250x_{2}=2250 by subtracting like terms on each side of the equal sign.

$$7500x_{1}-7500x_{1}+2250x_{2}+4500x_{2}=2250-2000$$

Add 7500x_{1} to -7500x_{1}. Terms 7500x_{1} and -7500x_{1} cancel out, leaving an equation with only one variable that can be solved.

$$2250x_{2}+4500x_{2}=2250-2000$$

Add 2250x_{2} to 4500x_{2}.

$$6750x_{2}=2250-2000$$

Add 2250 to -2000.

$$6750x_{2}=250$$

Divide both sides by 6750.

$$x_{2}=\frac{1}{27}$$

Substitute \frac{1}{27} for x_{2} in 75x_{1}-45x_{2}=20. Because the resulting equation contains only one variable, you can solve for x_{1} directly.

$$75x_{1}-45\times \frac{1}{27}=20$$

Multiply -45 times \frac{1}{27}.

$$75x_{1}-\frac{5}{3}=20$$

Add \frac{5}{3} to both sides of the equation.

$$75x_{1}=\frac{65}{3}$$

Divide both sides by 75.

$$x_{1}=\frac{13}{45}$$

The system is now solved.

$$x_{1}=\frac{13}{45},x_{2}=\frac{1}{27}$$