Solve {l}{x+y=60}{14x=20y}

$$\left. \begin{array} { l } { x + y = 60 } \\ { 14 x = 20 y } \end{array} \right.$$

Solve for x, y

$$x = \frac{600}{17} = 35\frac{5}{17} \approx 35.294117647$$

$$y = \frac{420}{17} = 24\frac{12}{17} \approx 24.705882353$$

Steps Using Substitution

Consider the second equation. Subtract 20y from both sides.

$$14x-20y=0$$

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.

$$x+y=60,14x-20y=0$$

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

$$x+y=60$$

Subtract y from both sides of the equation.

$$x=-y+60$$

Substitute -y+60 for x in the other equation, 14x-20y=0.

$$14\left(-y+60\right)-20y=0$$

Multiply 14 times -y+60.

$$-14y+840-20y=0$$

Add -14y to -20y.

$$-34y+840=0$$

Subtract 840 from both sides of the equation.

$$-34y=-840$$

Divide both sides by -34.

$$y=\frac{420}{17}$$

Substitute \frac{420}{17} for y in x=-y+60. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{420}{17}+60$$

Add 60 to -\frac{420}{17}.

$$x=\frac{600}{17}$$

The system is now solved.

$$x=\frac{600}{17},y=\frac{420}{17}$$

Steps Using Matrices

Consider the second equation. Subtract 20y from both sides.

$$14x-20y=0$$

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

$$x+y=60,14x-20y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\14&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\14&-20\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\14&-20\end{matrix}\right))\left(\begin{matrix}1&1\\14&-20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\14&-20\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\14&-20\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\14&-20\end{matrix}\right))\left(\begin{matrix}60\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{-20-14}&-\frac{1}{-20-14}\\-\frac{14}{-20-14}&\frac{1}{-20-14}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{17}&\frac{1}{34}\\\frac{7}{17}&-\frac{1}{34}\end{matrix}\right)\left(\begin{matrix}60\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{17}\times 60\\\frac{7}{17}\times 60\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{600}{17}\\\frac{420}{17}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{600}{17},y=\frac{420}{17}$$

Steps Using Elimination

Consider the second equation. Subtract 20y from both sides.

$$14x-20y=0$$

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.

$$x+y=60,14x-20y=0$$

To make x and 14x equal, multiply all terms on each side of the first equation by 14 and all terms on each side of the second by 1.

$$14x+14y=14\times 60,14x-20y=0$$

Simplify.

$$14x+14y=840,14x-20y=0$$

Subtract 14x-20y=0 from 14x+14y=840 by subtracting like terms on each side of the equal sign.

$$14x-14x+14y+20y=840$$

Add 14x to -14x. Terms 14x and -14x cancel out, leaving an equation with only one variable that can be solved.

$$14y+20y=840$$

Add 14y to 20y.

$$34y=840$$

Divide both sides by 34.

$$y=\frac{420}{17}$$

Substitute \frac{420}{17} for y in 14x-20y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$14x-20\times \frac{420}{17}=0$$

Multiply -20 times \frac{420}{17}.

$$14x-\frac{8400}{17}=0$$

Add \frac{8400}{17} to both sides of the equation.

$$14x=\frac{8400}{17}$$

Divide both sides by 14.

$$x=\frac{600}{17}$$

The system is now solved.

$$x=\frac{600}{17},y=\frac{420}{17}$$