Solve {l}{20x=(50+x+y)}{50y=5(50+x+y)}

$$\left. \begin{array} { l } { 20 x = ( 50 + x + y ) } \\ { 50 y = 5 ( 50 + x + y ) } \end{array} \right.$$

Solve for x, y

$$x = \frac{50}{17} = 2\frac{16}{17} \approx 2.941176471$$

$$y = \frac{100}{17} = 5\frac{15}{17} \approx 5.882352941$$

Steps Using Substitution

Consider the first equation. Subtract x from both sides.

$$20x-x=50+y$$

Combine 20x and -x to get 19x.

$$19x=50+y$$

Subtract y from both sides.

$$19x-y=50$$

Consider the second equation. Use the distributive property to multiply 5 by 50+x+y.

$$50y=250+5x+5y$$

Subtract 5x from both sides.

$$50y-5x=250+5y$$

Subtract 5y from both sides.

$$50y-5x-5y=250$$

Combine 50y and -5y to get 45y.

$$45y-5x=250$$

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.

$$19x-y=50,-5x+45y=250$$

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

$$19x-y=50$$

Add y to both sides of the equation.

$$19x=y+50$$

Divide both sides by 19.

$$x=\frac{1}{19}\left(y+50\right)$$

Multiply \frac{1}{19} times y+50.

$$x=\frac{1}{19}y+\frac{50}{19}$$

Substitute \frac{50+y}{19} for x in the other equation, -5x+45y=250.

$$-5\left(\frac{1}{19}y+\frac{50}{19}\right)+45y=250$$

Multiply -5 times \frac{50+y}{19}.

$$-\frac{5}{19}y-\frac{250}{19}+45y=250$$

Add -\frac{5y}{19} to 45y.

$$\frac{850}{19}y-\frac{250}{19}=250$$

Add \frac{250}{19} to both sides of the equation.

$$\frac{850}{19}y=\frac{5000}{19}$$

Divide both sides of the equation by \frac{850}{19}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{100}{17}$$

Substitute \frac{100}{17} for y in x=\frac{1}{19}y+\frac{50}{19}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{19}\times \frac{100}{17}+\frac{50}{19}$$

Multiply \frac{1}{19} times \frac{100}{17} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{100}{323}+\frac{50}{19}$$

Add \frac{50}{19} to \frac{100}{323} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{50}{17}$$

The system is now solved.

$$x=\frac{50}{17},y=\frac{100}{17}$$

Steps Using Matrices

Consider the first equation. Subtract x from both sides.

$$20x-x=50+y$$

Combine 20x and -x to get 19x.

$$19x=50+y$$

Subtract y from both sides.

$$19x-y=50$$

Consider the second equation. Use the distributive property to multiply 5 by 50+x+y.

$$50y=250+5x+5y$$

Subtract 5x from both sides.

$$50y-5x=250+5y$$

Subtract 5y from both sides.

$$50y-5x-5y=250$$

Combine 50y and -5y to get 45y.

$$45y-5x=250$$

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

$$19x-y=50,-5x+45y=250$$

Write the equations in matrix form.

$$\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\250\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}19&-1\\-5&45\end{matrix}\right).

$$inverse(\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right))\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right))\left(\begin{matrix}50\\250\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right))\left(\begin{matrix}50\\250\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}19&-1\\-5&45\end{matrix}\right))\left(\begin{matrix}50\\250\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{45}{19\times 45-\left(-\left(-5\right)\right)}&-\frac{-1}{19\times 45-\left(-\left(-5\right)\right)}\\-\frac{-5}{19\times 45-\left(-\left(-5\right)\right)}&\frac{19}{19\times 45-\left(-\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}50\\250\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{170}&\frac{1}{850}\\\frac{1}{170}&\frac{19}{850}\end{matrix}\right)\left(\begin{matrix}50\\250\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{170}\times 50+\frac{1}{850}\times 250\\\frac{1}{170}\times 50+\frac{19}{850}\times 250\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{50}{17}\\\frac{100}{17}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{50}{17},y=\frac{100}{17}$$

Steps Using Elimination

Consider the first equation. Subtract x from both sides.

$$20x-x=50+y$$

Combine 20x and -x to get 19x.

$$19x=50+y$$

Subtract y from both sides.

$$19x-y=50$$

Consider the second equation. Use the distributive property to multiply 5 by 50+x+y.

$$50y=250+5x+5y$$

Subtract 5x from both sides.

$$50y-5x=250+5y$$

Subtract 5y from both sides.

$$50y-5x-5y=250$$

Combine 50y and -5y to get 45y.

$$45y-5x=250$$

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.

$$19x-y=50,-5x+45y=250$$

To make 19x and -5x equal, multiply all terms on each side of the first equation by -5 and all terms on each side of the second by 19.

$$-5\times 19x-5\left(-1\right)y=-5\times 50,19\left(-5\right)x+19\times 45y=19\times 250$$

Simplify.

$$-95x+5y=-250,-95x+855y=4750$$

Subtract -95x+855y=4750 from -95x+5y=-250 by subtracting like terms on each side of the equal sign.

$$-95x+95x+5y-855y=-250-4750$$

Add -95x to 95x. Terms -95x and 95x cancel out, leaving an equation with only one variable that can be solved.

$$5y-855y=-250-4750$$

Add 5y to -855y.

$$-850y=-250-4750$$

Add -250 to -4750.

$$-850y=-5000$$

Divide both sides by -850.

$$y=\frac{100}{17}$$

Substitute \frac{100}{17} for y in -5x+45y=250. Because the resulting equation contains only one variable, you can solve for x directly.

$$-5x+45\times \frac{100}{17}=250$$

Multiply 45 times \frac{100}{17}.

$$-5x+\frac{4500}{17}=250$$

Subtract \frac{4500}{17} from both sides of the equation.

$$-5x=-\frac{250}{17}$$

Divide both sides by -5.

$$x=\frac{50}{17}$$

The system is now solved.

$$x=\frac{50}{17},y=\frac{100}{17}$$