Solve {l}{3x+y=5}{x+2/5+y/2=-1}

$$\left\{ \begin{array} { l } { 3 x + y = 5 } \\ { \frac { x + 2 } { 5 } + \frac { y } { 2 } = - 1 } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=-4$$

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.

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

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

$$3x+y=5$$

Subtract y from both sides of the equation.

$$3x=-y+5$$

Divide both sides by 3.

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

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

$$x=-\frac{1}{3}y+\frac{5}{3}$$

Substitute \frac{-y+5}{3} for x in the other equation, \frac{1}{5}\left(x+2\right)+\frac{1}{2}y=-1.

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

Add \frac{5}{3} to 2.

$$\frac{1}{5}\left(-\frac{1}{3}y+\frac{11}{3}\right)+\frac{1}{2}y=-1$$

Multiply \frac{1}{5} times \frac{-y+11}{3}.

$$-\frac{1}{15}y+\frac{11}{15}+\frac{1}{2}y=-1$$

Add -\frac{y}{15} to \frac{y}{2}.

$$\frac{13}{30}y+\frac{11}{15}=-1$$

Subtract \frac{11}{15} from both sides of the equation.

$$\frac{13}{30}y=-\frac{26}{15}$$

Divide both sides of the equation by \frac{13}{30}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-4$$

Substitute -4 for y in x=-\frac{1}{3}y+\frac{5}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{3}\left(-4\right)+\frac{5}{3}$$

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

$$x=\frac{4+5}{3}$$

Add \frac{5}{3} to \frac{4}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=3$$

The system is now solved.

$$x=3,y=-4$$

Steps Using Matrices

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

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

Simplify the second equation to put it in standard form.

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

Multiply \frac{1}{5} times x+2.

$$\frac{1}{5}x+\frac{2}{5}+\frac{1}{2}y=-1$$

Subtract \frac{2}{5} from both sides of the equation.

$$\frac{1}{5}x+\frac{1}{2}y=-\frac{7}{5}$$

Write the equations in matrix form.

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

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&1\\\frac{1}{5}&\frac{1}{2}\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\\frac{1}{5}&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}5\\-\frac{7}{5}\end{matrix}\right)$$

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

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

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.

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

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}&-\frac{10}{13}\\-\frac{2}{13}&\frac{30}{13}\end{matrix}\right)\left(\begin{matrix}5\\-\frac{7}{5}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{13}\times 5-\frac{10}{13}\left(-\frac{7}{5}\right)\\-\frac{2}{13}\times 5+\frac{30}{13}\left(-\frac{7}{5}\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=3,y=-4$$