Solve {l}{9x+my+3=0}{mx+4y+2=0}

$$\left. \begin{array} { l } { 9 x + m y + 3 = 0 } \\ { m x + 4 y + 2 = 0 } \end{array} \right.$$

Solve for x, y (complex solution)

$$\left\{\begin{matrix}x=-\frac{2}{m+6}\text{, }y=-\frac{3}{m+6}\text{, }&m\neq -6\\x=\frac{-2y-1}{3}\text{, }y\in \mathrm{C}\text{, }&m=6\end{matrix}\right.$$

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.

$$9x+my+3=0,mx+4y+2=0$$

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

$$9x+my+3=0$$

Subtract 3 from both sides of the equation.

$$9x+my=-3$$

Subtract my from both sides of the equation.

$$9x=\left(-m\right)y-3$$

Divide both sides by 9.

$$x=\frac{1}{9}\left(\left(-m\right)y-3\right)$$

Multiply \frac{1}{9} times -my-3.

$$x=\left(-\frac{m}{9}\right)y-\frac{1}{3}$$

Substitute -\frac{my}{9}-\frac{1}{3} for x in the other equation, mx+4y+2=0.

$$m\left(\left(-\frac{m}{9}\right)y-\frac{1}{3}\right)+4y+2=0$$

Multiply m times -\frac{my}{9}-\frac{1}{3}.

$$\left(-\frac{m^{2}}{9}\right)y-\frac{m}{3}+4y+2=0$$

Add -\frac{m^{2}y}{9} to 4y.

$$\left(-\frac{m^{2}}{9}+4\right)y-\frac{m}{3}+2=0$$

Subtract -\frac{m}{3}+2 from both sides of the equation.

$$\left(-\frac{m^{2}}{9}+4\right)y=\frac{m}{3}-2$$

Divide both sides by -\frac{m^{2}}{9}+4.

$$y=-\frac{3}{m+6}$$

Substitute -\frac{3}{6+m} for y in x=\left(-\frac{m}{9}\right)y-\frac{1}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-\frac{m}{9}\right)\left(-\frac{3}{m+6}\right)-\frac{1}{3}$$

Multiply -\frac{m}{9} times -\frac{3}{6+m}.

$$x=\frac{m}{3\left(m+6\right)}-\frac{1}{3}$$

Add -\frac{1}{3} to \frac{m}{3\left(6+m\right)}.

$$x=-\frac{2}{m+6}$$

The system is now solved.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

Steps Using Matrices

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

$$9x+my+3=0,mx+4y+2=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&m\\m&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-mm}&-\frac{m}{9\times 4-mm}\\-\frac{m}{9\times 4-mm}&\frac{9}{9\times 4-mm}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}&-\frac{m}{36-m^{2}}\\-\frac{m}{36-m^{2}}&\frac{9}{36-m^{2}}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}\left(-3\right)+\left(-\frac{m}{36-m^{2}}\right)\left(-2\right)\\\left(-\frac{m}{36-m^{2}}\right)\left(-3\right)+\frac{9}{36-m^{2}}\left(-2\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{m+6}\\-\frac{3}{m+6}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

To make 9x and mx equal, multiply all terms on each side of the first equation by m and all terms on each side of the second by 9.

$$m\times 9x+mmy+m\times 3=0,9mx+9\times 4y+9\times 2=0$$

Simplify.

$$9mx+m^{2}y+3m=0,9mx+36y+18=0$$

Subtract 9mx+36y+18=0 from 9mx+m^{2}y+3m=0 by subtracting like terms on each side of the equal sign.

$$9mx+\left(-9m\right)x+m^{2}y-36y+3m-18=0$$

Add 9mx to -9mx. Terms 9mx and -9mx cancel out, leaving an equation with only one variable that can be solved.

$$m^{2}y-36y+3m-18=0$$

Add m^{2}y to -36y.

$$\left(m^{2}-36\right)y+3m-18=0$$

Subtract -18+3m from both sides of the equation.

$$\left(m^{2}-36\right)y=18-3m$$

Divide both sides by m^{2}-36.

$$y=-\frac{3}{m+6}$$

Substitute -\frac{3}{6+m} for y in mx+4y+2=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$mx+4\left(-\frac{3}{m+6}\right)+2=0$$

Multiply 4 times -\frac{3}{6+m}.

$$mx-\frac{12}{m+6}+2=0$$

Add -\frac{12}{6+m} to 2.

$$mx+\frac{2m}{m+6}=0$$

Subtract \frac{2m}{6+m} from both sides of the equation.

$$mx=-\frac{2m}{m+6}$$

Divide both sides by m.

$$x=-\frac{2}{m+6}$$

The system is now solved.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

Solve for x, y

$$\left\{\begin{matrix}x=-\frac{2}{m+6}\text{, }y=-\frac{3}{m+6}\text{, }&|m|\neq 6\\x=\frac{-2y-1}{3}\text{, }y\in \mathrm{R}\text{, }&m=6\end{matrix}\right.$$

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.

$$9x+my+3=0,mx+4y+2=0$$

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

$$9x+my+3=0$$

Subtract 3 from both sides of the equation.

$$9x+my=-3$$

Subtract my from both sides of the equation.

$$9x=\left(-m\right)y-3$$

Divide both sides by 9.

$$x=\frac{1}{9}\left(\left(-m\right)y-3\right)$$

Multiply \frac{1}{9} times -my-3.

$$x=\left(-\frac{m}{9}\right)y-\frac{1}{3}$$

Substitute -\frac{my}{9}-\frac{1}{3} for x in the other equation, mx+4y+2=0.

$$m\left(\left(-\frac{m}{9}\right)y-\frac{1}{3}\right)+4y+2=0$$

Multiply m times -\frac{my}{9}-\frac{1}{3}.

$$\left(-\frac{m^{2}}{9}\right)y-\frac{m}{3}+4y+2=0$$

Add -\frac{m^{2}y}{9} to 4y.

$$\left(-\frac{m^{2}}{9}+4\right)y-\frac{m}{3}+2=0$$

Subtract -\frac{m}{3}+2 from both sides of the equation.

$$\left(-\frac{m^{2}}{9}+4\right)y=\frac{m}{3}-2$$

Divide both sides by -\frac{m^{2}}{9}+4.

$$y=-\frac{3}{m+6}$$

Substitute -\frac{3}{6+m} for y in x=\left(-\frac{m}{9}\right)y-\frac{1}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-\frac{m}{9}\right)\left(-\frac{3}{m+6}\right)-\frac{1}{3}$$

Multiply -\frac{m}{9} times -\frac{3}{6+m}.

$$x=\frac{m}{3\left(m+6\right)}-\frac{1}{3}$$

Add -\frac{1}{3} to \frac{m}{3\left(6+m\right)}.

$$x=-\frac{2}{m+6}$$

The system is now solved.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

Steps Using Matrices

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

$$9x+my+3=0,mx+4y+2=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&m\\m&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}9&m\\m&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&m\\m&4\end{matrix}\right))\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{9\times 4-mm}&-\frac{m}{9\times 4-mm}\\-\frac{m}{9\times 4-mm}&\frac{9}{9\times 4-mm}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}&-\frac{m}{36-m^{2}}\\-\frac{m}{36-m^{2}}&\frac{9}{36-m^{2}}\end{matrix}\right)\left(\begin{matrix}-3\\-2\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{36-m^{2}}\left(-3\right)+\left(-\frac{m}{36-m^{2}}\right)\left(-2\right)\\\left(-\frac{m}{36-m^{2}}\right)\left(-3\right)+\frac{9}{36-m^{2}}\left(-2\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{m+6}\\-\frac{3}{m+6}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$

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.

$$9x+my+3=0,mx+4y+2=0$$

To make 9x and mx equal, multiply all terms on each side of the first equation by m and all terms on each side of the second by 9.

$$m\times 9x+mmy+m\times 3=0,9mx+9\times 4y+9\times 2=0$$

Simplify.

$$9mx+m^{2}y+3m=0,9mx+36y+18=0$$

Subtract 9mx+36y+18=0 from 9mx+m^{2}y+3m=0 by subtracting like terms on each side of the equal sign.

$$9mx+\left(-9m\right)x+m^{2}y-36y+3m-18=0$$

Add 9mx to -9mx. Terms 9mx and -9mx cancel out, leaving an equation with only one variable that can be solved.

$$m^{2}y-36y+3m-18=0$$

Add m^{2}y to -36y.

$$\left(m^{2}-36\right)y+3m-18=0$$

Subtract -18+3m from both sides of the equation.

$$\left(m^{2}-36\right)y=18-3m$$

Divide both sides by m^{2}-36.

$$y=-\frac{3}{m+6}$$

Substitute -\frac{3}{6+m} for y in mx+4y+2=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$mx+4\left(-\frac{3}{m+6}\right)+2=0$$

Multiply 4 times -\frac{3}{6+m}.

$$mx-\frac{12}{m+6}+2=0$$

Add -\frac{12}{6+m} to 2.

$$mx+\frac{2m}{m+6}=0$$

Subtract \frac{2m}{6+m} from both sides of the equation.

$$mx=-\frac{2m}{m+6}$$

Divide both sides by m.

$$x=-\frac{2}{m+6}$$

The system is now solved.

$$x=-\frac{2}{m+6},y=-\frac{3}{m+6}$$