Solve {l}{60x+20y=4100}{x+y=105}

$$\left. \begin{array} { l } { 60 x + 20 y = 4100 } \\ { x + y = 105 } \end{array} \right.$$

Solve for x, y

$$x=50$$

$$y=55$$

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.

$$60x+20y=4100,x+y=105$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$60x+20y=4100$$

Subtract 20y from both sides of the equation.

$$60x=-20y+4100$$

Divide both sides by 60.

$$x=\frac{1}{60}\left(-20y+4100\right)$$

Multiply \frac{1}{60} times -20y+4100.

$$x=-\frac{1}{3}y+\frac{205}{3}$$

Substitute \frac{-y+205}{3} for x in the other equation, x+y=105.

$$-\frac{1}{3}y+\frac{205}{3}+y=105$$

Add -\frac{y}{3} to y.

$$\frac{2}{3}y+\frac{205}{3}=105$$

Subtract \frac{205}{3} from both sides of the equation.

$$\frac{2}{3}y=\frac{110}{3}$$

Divide both sides of the equation by \frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=55$$

Substitute 55 for y in x=-\frac{1}{3}y+\frac{205}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{3}\times 55+\frac{205}{3}$$

Multiply -\frac{1}{3} times 55.

$$x=\frac{-55+205}{3}$$

Add \frac{205}{3} to -\frac{55}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=50$$

The system is now solved.

$$x=50,y=55$$

Steps Using Matrices

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

$$60x+20y=4100,x+y=105$$

Write the equations in matrix form.

$$\left(\begin{matrix}60&20\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4100\\105\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}60&20\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}60&20\\1&1\end{matrix}\right))\left(\begin{matrix}60&20\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&20\\1&1\end{matrix}\right))\left(\begin{matrix}4100\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&20\\1&1\end{matrix}\right))\left(\begin{matrix}4100\\105\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&20\\1&1\end{matrix}\right))\left(\begin{matrix}4100\\105\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{60-20}&-\frac{20}{60-20}\\-\frac{1}{60-20}&\frac{60}{60-20}\end{matrix}\right)\left(\begin{matrix}4100\\105\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{40}&-\frac{1}{2}\\-\frac{1}{40}&\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}4100\\105\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{40}\times 4100-\frac{1}{2}\times 105\\-\frac{1}{40}\times 4100+\frac{3}{2}\times 105\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\55\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=50,y=55$$

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.

$$60x+20y=4100,x+y=105$$

To make 60x and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 60.

$$60x+20y=4100,60x+60y=60\times 105$$

Simplify.

$$60x+20y=4100,60x+60y=6300$$

Subtract 60x+60y=6300 from 60x+20y=4100 by subtracting like terms on each side of the equal sign.

$$60x-60x+20y-60y=4100-6300$$

Add 60x to -60x. Terms 60x and -60x cancel out, leaving an equation with only one variable that can be solved.

$$20y-60y=4100-6300$$

Add 20y to -60y.

$$-40y=4100-6300$$

Add 4100 to -6300.

$$-40y=-2200$$

Divide both sides by -40.

$$y=55$$

Substitute 55 for y in x+y=105. Because the resulting equation contains only one variable, you can solve for x directly.

$$x+55=105$$

Subtract 55 from both sides of the equation.

$$x=50$$

The system is now solved.

$$x=50,y=55$$