Solve {l}{-2x+6y=723}{5x+12y=55}

$$\left. \begin{array} { l } { - 2 x + 6 y = 723 } \\ { 5 x + 12 y = 55 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{1391}{9} = -154\frac{5}{9} \approx -154.555555556$$

$$y = \frac{3725}{54} = 68\frac{53}{54} \approx 68.981481481$$

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.

$$-2x+6y=723,5x+12y=55$$

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

$$-2x+6y=723$$

Subtract 6y from both sides of the equation.

$$-2x=-6y+723$$

Divide both sides by -2.

$$x=-\frac{1}{2}\left(-6y+723\right)$$

Multiply -\frac{1}{2} times -6y+723.

$$x=3y-\frac{723}{2}$$

Substitute 3y-\frac{723}{2} for x in the other equation, 5x+12y=55.

$$5\left(3y-\frac{723}{2}\right)+12y=55$$

Multiply 5 times 3y-\frac{723}{2}.

$$15y-\frac{3615}{2}+12y=55$$

Add 15y to 12y.

$$27y-\frac{3615}{2}=55$$

Add \frac{3615}{2} to both sides of the equation.

$$27y=\frac{3725}{2}$$

Divide both sides by 27.

$$y=\frac{3725}{54}$$

Substitute \frac{3725}{54} for y in x=3y-\frac{723}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=3\times \frac{3725}{54}-\frac{723}{2}$$

Multiply 3 times \frac{3725}{54}.

$$x=\frac{3725}{18}-\frac{723}{2}$$

Add -\frac{723}{2} to \frac{3725}{18} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{1391}{9}$$

The system is now solved.

$$x=-\frac{1391}{9},y=\frac{3725}{54}$$

Steps Using Matrices

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

$$-2x+6y=723,5x+12y=55$$

Write the equations in matrix form.

$$\left(\begin{matrix}-2&6\\5&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}723\\55\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-2&6\\5&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}-2&6\\5&12\end{matrix}\right))\left(\begin{matrix}-2&6\\5&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&6\\5&12\end{matrix}\right))\left(\begin{matrix}723\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&6\\5&12\end{matrix}\right))\left(\begin{matrix}723\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-2&6\\5&12\end{matrix}\right))\left(\begin{matrix}723\\55\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{-2\times 12-6\times 5}&-\frac{6}{-2\times 12-6\times 5}\\-\frac{5}{-2\times 12-6\times 5}&-\frac{2}{-2\times 12-6\times 5}\end{matrix}\right)\left(\begin{matrix}723\\55\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}&\frac{1}{9}\\\frac{5}{54}&\frac{1}{27}\end{matrix}\right)\left(\begin{matrix}723\\55\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{9}\times 723+\frac{1}{9}\times 55\\\frac{5}{54}\times 723+\frac{1}{27}\times 55\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1391}{9}\\\frac{3725}{54}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{1391}{9},y=\frac{3725}{54}$$

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.

$$-2x+6y=723,5x+12y=55$$

To make -2x 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 -2.

$$5\left(-2\right)x+5\times 6y=5\times 723,-2\times 5x-2\times 12y=-2\times 55$$

Simplify.

$$-10x+30y=3615,-10x-24y=-110$$

Subtract -10x-24y=-110 from -10x+30y=3615 by subtracting like terms on each side of the equal sign.

$$-10x+10x+30y+24y=3615+110$$

Add -10x to 10x. Terms -10x and 10x cancel out, leaving an equation with only one variable that can be solved.

$$30y+24y=3615+110$$

Add 30y to 24y.

$$54y=3615+110$$

Add 3615 to 110.

$$54y=3725$$

Divide both sides by 54.

$$y=\frac{3725}{54}$$

Substitute \frac{3725}{54} for y in 5x+12y=55. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x+12\times \frac{3725}{54}=55$$

Multiply 12 times \frac{3725}{54}.

$$5x+\frac{7450}{9}=55$$

Subtract \frac{7450}{9} from both sides of the equation.

$$5x=-\frac{6955}{9}$$

Divide both sides by 5.

$$x=-\frac{1391}{9}$$

The system is now solved.

$$x=-\frac{1391}{9},y=\frac{3725}{54}$$