Solve {l}{2x-3y=-5m}{4x+y=11m}

$$\left\{ \begin{array} { l } { 2 x - 3 y = - 5 m } \\ { 4 x + y = 11 m } \end{array} \right.$$

Solve for x, y

$$x=2m$$

$$y=3m$$

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.

$$2x-3y=-5m,4x+y=11m$$

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

$$2x-3y=-5m$$

Add 3y to both sides of the equation.

$$2x=3y-5m$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(3y-5m\right)$$

Multiply \frac{1}{2} times 3y-5m.

$$x=\frac{3}{2}y-\frac{5m}{2}$$

Substitute \frac{3y-5m}{2} for x in the other equation, 4x+y=11m.

$$4\left(\frac{3}{2}y-\frac{5m}{2}\right)+y=11m$$

Multiply 4 times \frac{3y-5m}{2}.

$$6y-10m+y=11m$$

Add 6y to y.

$$7y-10m=11m$$

Add 10m to both sides of the equation.

$$7y=21m$$

Divide both sides by 7.

$$y=3m$$

Substitute 3m for y in x=\frac{3}{2}y-\frac{5m}{2}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{2}\times 3m-\frac{5m}{2}$$

Multiply \frac{3}{2} times 3m.

$$x=\frac{9m-5m}{2}$$

Add -\frac{5m}{2} to \frac{9m}{2}.

$$x=2m$$

The system is now solved.

$$x=2m,y=3m$$

Steps Using Matrices

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

$$2x-3y=-5m,4x+y=11m$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&-3\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&-3\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}2&-3\\4&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&1\end{matrix}\right))\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-3\times 4\right)}&-\frac{-3}{2-\left(-3\times 4\right)}\\-\frac{4}{2-\left(-3\times 4\right)}&\frac{2}{2-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}&\frac{3}{14}\\-\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-5m\\11m\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{14}\left(-5m\right)+\frac{3}{14}\times 11m\\-\frac{2}{7}\left(-5m\right)+\frac{1}{7}\times 11m\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=2m,y=3m$$

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.

$$2x-3y=-5m,4x+y=11m$$

To make 2x and 4x equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 2.

$$4\times 2x+4\left(-3\right)y=4\left(-5m\right),2\times 4x+2y=2\times 11m$$

Simplify.

$$8x-12y=-20m,8x+2y=22m$$

Subtract 8x+2y=22m from 8x-12y=-20m by subtracting like terms on each side of the equal sign.

$$8x-8x-12y-2y=-20m-22m$$

Add 8x to -8x. Terms 8x and -8x cancel out, leaving an equation with only one variable that can be solved.

$$-12y-2y=-20m-22m$$

Add -12y to -2y.

$$-14y=-20m-22m$$

Add -20m to -22m.

$$-14y=-42m$$

Divide both sides by -14.

$$y=3m$$

Substitute 3m for y in 4x+y=11m. Because the resulting equation contains only one variable, you can solve for x directly.

$$4x+3m=11m$$

Subtract 3m from both sides of the equation.

$$4x=8m$$

Divide both sides by 4.

$$x=2m$$

The system is now solved.

$$x=2m,y=3m$$