Solve {l}{60x+60y=1170-300}{70x+140y=2035}

$$\left. \begin{array}{l}{ 60 x + 60 y = 1170 - 300 }\\{ 70 x + 140 y = 2035 }\end{array} \right.$$

Solve for x, y

$$x=-\frac{1}{14}\approx -0.071428571$$

$$y = \frac{102}{7} = 14\frac{4}{7} \approx 14.571428571$$

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.

$$60x+60y=870,70x+140y=2035$$

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

$$60x+60y=870$$

Subtract 60y from both sides of the equation.

$$60x=-60y+870$$

Divide both sides by 60.

$$x=\frac{1}{60}\left(-60y+870\right)$$

Multiply \frac{1}{60} times -60y+870.

$$x=-y+\frac{29}{2}$$

Substitute -y+\frac{29}{2} for x in the other equation, 70x+140y=2035.

$$70\left(-y+\frac{29}{2}\right)+140y=2035$$

Multiply 70 times -y+\frac{29}{2}.

$$-70y+1015+140y=2035$$

Add -70y to 140y.

$$70y+1015=2035$$

Subtract 1015 from both sides of the equation.

$$70y=1020$$

Divide both sides by 70.

$$y=\frac{102}{7}$$

Substitute \frac{102}{7} for y in x=-y+\frac{29}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{102}{7}+\frac{29}{2}$$

Add \frac{29}{2} to -\frac{102}{7} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{1}{14}$$

The system is now solved.

$$x=-\frac{1}{14},y=\frac{102}{7}$$

Steps Using Matrices

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

$$60x+60y=870,70x+140y=2035$$

Write the equations in matrix form.

$$\left(\begin{matrix}60&60\\70&140\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}60&60\\70&140\end{matrix}\right).

$$inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}60&60\\70&140\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}60&60\\70&140\end{matrix}\right))\left(\begin{matrix}870\\2035\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{140}{60\times 140-60\times 70}&-\frac{60}{60\times 140-60\times 70}\\-\frac{70}{60\times 140-60\times 70}&\frac{60}{60\times 140-60\times 70}\end{matrix}\right)\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}&-\frac{1}{70}\\-\frac{1}{60}&\frac{1}{70}\end{matrix}\right)\left(\begin{matrix}870\\2035\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{30}\times 870-\frac{1}{70}\times 2035\\-\frac{1}{60}\times 870+\frac{1}{70}\times 2035\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\\\frac{102}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{1}{14},y=\frac{102}{7}$$

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.

$$60x+60y=870,70x+140y=2035$$

To make 60x and 70x equal, multiply all terms on each side of the first equation by 70 and all terms on each side of the second by 60.

$$70\times 60x+70\times 60y=70\times 870,60\times 70x+60\times 140y=60\times 2035$$

Simplify.

$$4200x+4200y=60900,4200x+8400y=122100$$

Subtract 4200x+8400y=122100 from 4200x+4200y=60900 by subtracting like terms on each side of the equal sign.

$$4200x-4200x+4200y-8400y=60900-122100$$

Add 4200x to -4200x. Terms 4200x and -4200x cancel out, leaving an equation with only one variable that can be solved.

$$4200y-8400y=60900-122100$$

Add 4200y to -8400y.

$$-4200y=60900-122100$$

Add 60900 to -122100.

$$-4200y=-61200$$

Divide both sides by -4200.

$$y=\frac{102}{7}$$

Substitute \frac{102}{7} for y in 70x+140y=2035. Because the resulting equation contains only one variable, you can solve for x directly.

$$70x+140\times \frac{102}{7}=2035$$

Multiply 140 times \frac{102}{7}.

$$70x+2040=2035$$

Subtract 2040 from both sides of the equation.

$$70x=-5$$

Divide both sides by 70.

$$x=-\frac{1}{14}$$

The system is now solved.

$$x=-\frac{1}{14},y=\frac{102}{7}$$