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

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

Solve for x, y

$$x=2$$

$$y=3$$

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.

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

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

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

Subtract \frac{y}{3} from both sides of the equation.

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

Multiply both sides by 2.

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

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

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

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

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

Add 4 to 3.

$$2\left(-\frac{2}{3}y+7\right)-3y=1$$

Multiply 2 times -\frac{2y}{3}+7.

$$-\frac{4}{3}y+14-3y=1$$

Add -\frac{4y}{3} to -3y.

$$-\frac{13}{3}y+14=1$$

Subtract 14 from both sides of the equation.

$$-\frac{13}{3}y=-13$$

Divide both sides of the equation by -\frac{13}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=3$$

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

$$x=-\frac{2}{3}\times 3+4$$

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

$$x=-2+4$$

Add 4 to -2.

$$x=2$$

The system is now solved.

$$x=2,y=3$$

Steps Using Matrices

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

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

Simplify the second equation to put it in standard form.

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

Multiply 2 times x+3.

$$2x+6-3y=1$$

Subtract 6 from both sides of the equation.

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

Write the equations in matrix form.

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

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

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

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

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

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

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

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.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}&\frac{2}{13}\\\frac{12}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}2\\-5\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}\times 2+\frac{2}{13}\left(-5\right)\\\frac{12}{13}\times 2-\frac{3}{13}\left(-5\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=2,y=3$$