Solve {l}{x-y=5}{x/2+y-7/5=-1}

$$\left\{ \begin{array} { l } { x - y = 5 } \\ { \frac { x } { 2 } + \frac { y - 7 } { 5 } = - 1 } \end{array} \right.$$

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.

$$x-y=5,\frac{1}{2}x+\frac{1}{5}\left(y-7\right)=-1$$

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

$$x-y=5$$

Add y to both sides of the equation.

$$x=y+5$$

Substitute y+5 for x in the other equation, \frac{1}{2}x+\frac{1}{5}\left(y-7\right)=-1.

$$\frac{1}{2}\left(y+5\right)+\frac{1}{5}\left(y-7\right)=-1$$

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

$$\frac{1}{2}y+\frac{5}{2}+\frac{1}{5}\left(y-7\right)=-1$$

Multiply \frac{1}{5} times y-7.

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

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

$$\frac{7}{10}y+\frac{5}{2}-\frac{7}{5}=-1$$

Add \frac{5}{2} to -\frac{7}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{7}{10}y+\frac{11}{10}=-1$$

Subtract \frac{11}{10} from both sides of the equation.

$$\frac{7}{10}y=-\frac{21}{10}$$

Divide both sides of the equation by \frac{7}{10}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-3$$

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

$$x=-3+5$$

Add 5 to -3.

$$x=2$$

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.

$$x-y=5,\frac{1}{2}x+\frac{1}{5}\left(y-7\right)=-1$$

Simplify the second equation to put it in standard form.

$$\frac{1}{2}x+\frac{1}{5}\left(y-7\right)=-1$$

Multiply \frac{1}{5} times y-7.

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

Add \frac{7}{5} to both sides of the equation.

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

Write the equations in matrix form.

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

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

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

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

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

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

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

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.

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

Do the arithmetic.

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

Multiply the matrices.

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

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.

$$x=2,y=-3$$