Solve {l}{2(2x-1)+3y=-11}{4x+3(1-3y)=30}

$$\left\{ \begin{array} { l } { 2 ( 2 x - 1 ) + 3 y = - 11 } \\ { 4 x + 3 ( 1 - 3 y ) = 30 } \end{array} \right.$$

Solve for x, y

$$x=0$$

$$y=-3$$

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.

$$2\left(2x-1\right)+3y=-11,4x+3\left(-3y+1\right)=30$$

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

$$2\left(2x-1\right)+3y=-11$$

Multiply 2 times 2x-1.

$$4x-2+3y=-11$$

Add 2 to both sides of the equation.

$$4x+3y=-9$$

Subtract 3y from both sides of the equation.

$$4x=-3y-9$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(-3y-9\right)$$

Multiply \frac{1}{4} times -3y-9.

$$x=-\frac{3}{4}y-\frac{9}{4}$$

Substitute \frac{-3y-9}{4} for x in the other equation, 4x+3\left(-3y+1\right)=30.

$$4\left(-\frac{3}{4}y-\frac{9}{4}\right)+3\left(-3y+1\right)=30$$

Multiply 4 times \frac{-3y-9}{4}.

$$-3y-9+3\left(-3y+1\right)=30$$

Multiply 3 times -3y+1.

$$-3y-9-9y+3=30$$

Add -3y to -9y.

$$-12y-9+3=30$$

Add -9 to 3.

$$-12y-6=30$$

Add 6 to both sides of the equation.

$$-12y=36$$

Divide both sides by -12.

$$y=-3$$

Substitute -3 for y in x=-\frac{3}{4}y-\frac{9}{4}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{4}\left(-3\right)-\frac{9}{4}$$

Multiply -\frac{3}{4} times -3.

$$x=\frac{9-9}{4}$$

Add -\frac{9}{4} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=0$$

The system is now solved.

$$x=0,y=-3$$

Steps Using Matrices

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

$$2\left(2x-1\right)+3y=-11,4x+3\left(-3y+1\right)=30$$

Simplify the first equation to put it in standard form.

$$2\left(2x-1\right)+3y=-11$$

Multiply 2 times 2x-1.

$$4x-2+3y=-11$$

Add 2 to both sides of the equation.

$$4x+3y=-9$$

Simplify the second equation to put it in standard form.

$$4x+3\left(-3y+1\right)=30$$

Multiply 3 times -3y+1.

$$4x-9y+3=30$$

Subtract 3 from both sides of the equation.

$$4x-9y=27$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&3\\4&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\27\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&3\\4&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\4&-9\end{matrix}\right))\left(\begin{matrix}4&3\\4&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\4&-9\end{matrix}\right))\left(\begin{matrix}-9\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\4&-9\end{matrix}\right))\left(\begin{matrix}-9\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\4&-9\end{matrix}\right))\left(\begin{matrix}-9\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{4\left(-9\right)-3\times 4}&-\frac{3}{4\left(-9\right)-3\times 4}\\-\frac{4}{4\left(-9\right)-3\times 4}&\frac{4}{4\left(-9\right)-3\times 4}\end{matrix}\right)\left(\begin{matrix}-9\\27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}&\frac{1}{16}\\\frac{1}{12}&-\frac{1}{12}\end{matrix}\right)\left(\begin{matrix}-9\\27\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{16}\left(-9\right)+\frac{1}{16}\times 27\\\frac{1}{12}\left(-9\right)-\frac{1}{12}\times 27\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=0,y=-3$$