Solve {l}{3(x-1)-2(y-2)=0}{x+3y=-4}

$$\left. \begin{array} { l } { 3 ( x - 1 ) - 2 ( y - 2 ) = 0 } \\ { x + 3 y = - 4 } \end{array} \right.$$

Solve for x, y

$$x=-1$$

$$y=-1$$

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.

$$3\left(x-1\right)-2\left(y-2\right)=0,x+3y=-4$$

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

$$3\left(x-1\right)-2\left(y-2\right)=0$$

Multiply 3 times x-1.

$$3x-3-2\left(y-2\right)=0$$

Multiply -2 times y-2.

$$3x-3-2y+4=0$$

Add -3 to 4.

$$3x-2y+1=0$$

Subtract 1 from both sides of the equation.

$$3x-2y=-1$$

Add 2y to both sides of the equation.

$$3x=2y-1$$

Divide both sides by 3.

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

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

$$x=\frac{2}{3}y-\frac{1}{3}$$

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

$$\frac{2}{3}y-\frac{1}{3}+3y=-4$$

Add \frac{2y}{3} to 3y.

$$\frac{11}{3}y-\frac{1}{3}=-4$$

Add \frac{1}{3} to both sides of the equation.

$$\frac{11}{3}y=-\frac{11}{3}$$

Divide both sides of the equation by \frac{11}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-1$$

Substitute -1 for y in x=\frac{2}{3}y-\frac{1}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

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

Multiply \frac{2}{3} times -1.

$$x=\frac{-2-1}{3}$$

Add -\frac{1}{3} to -\frac{2}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-1$$

The system is now solved.

$$x=-1,y=-1$$

Steps Using Matrices

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

$$3\left(x-1\right)-2\left(y-2\right)=0,x+3y=-4$$

Simplify the first equation to put it in standard form.

$$3\left(x-1\right)-2\left(y-2\right)=0$$

Multiply 3 times x-1.

$$3x-3-2\left(y-2\right)=0$$

Multiply -2 times y-2.

$$3x-3-2y+4=0$$

Add -3 to 4.

$$3x-2y+1=0$$

Subtract 1 from both sides of the equation.

$$3x-2y=-1$$

Write the equations in matrix form.

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

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

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

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

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

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

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

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.

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

Do the arithmetic.

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

Multiply the matrices.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-1,y=-1$$