Solve {l}{4x+7(y+10/11)=1}{5x-(y+10/11)=11}

$$\left\{ \begin{array} { l } { 4 x + 7 ( y + \frac { 10 } { 11 } ) = 1 } \\ { 5 x - ( y + \frac { 10 } { 11 } ) = 11 } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y = -\frac{21}{11} = -1\frac{10}{11} \approx -1.909090909$$

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.

$$4x+7\left(y+\frac{10}{11}\right)=1,5x-\left(y+\frac{10}{11}\right)=11$$

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

$$4x+7\left(y+\frac{10}{11}\right)=1$$

Multiply 7 times y+\frac{10}{11}.

$$4x+7y+\frac{70}{11}=1$$

Subtract \frac{70}{11} from both sides of the equation.

$$4x+7y=-\frac{59}{11}$$

Subtract 7y from both sides of the equation.

$$4x=-7y-\frac{59}{11}$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(-7y-\frac{59}{11}\right)$$

Multiply \frac{1}{4} times -7y-\frac{59}{11}.

$$x=-\frac{7}{4}y-\frac{59}{44}$$

Substitute -\frac{7y}{4}-\frac{59}{44} for x in the other equation, 5x-\left(y+\frac{10}{11}\right)=11.

$$5\left(-\frac{7}{4}y-\frac{59}{44}\right)-\left(y+\frac{10}{11}\right)=11$$

Multiply 5 times -\frac{7y}{4}-\frac{59}{44}.

$$-\frac{35}{4}y-\frac{295}{44}-\left(y+\frac{10}{11}\right)=11$$

Multiply -1 times y+\frac{10}{11}.

$$-\frac{35}{4}y-\frac{295}{44}-y-\frac{10}{11}=11$$

Add -\frac{35y}{4} to -y.

$$-\frac{39}{4}y-\frac{295}{44}-\frac{10}{11}=11$$

Add -\frac{295}{44} to -\frac{10}{11} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-\frac{39}{4}y-\frac{335}{44}=11$$

Add \frac{335}{44} to both sides of the equation.

$$-\frac{39}{4}y=\frac{819}{44}$$

Divide both sides of the equation by -\frac{39}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{21}{11}$$

Substitute -\frac{21}{11} for y in x=-\frac{7}{4}y-\frac{59}{44}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{7}{4}\left(-\frac{21}{11}\right)-\frac{59}{44}$$

Multiply -\frac{7}{4} times -\frac{21}{11} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{147-59}{44}$$

Add -\frac{59}{44} to \frac{147}{44} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=2$$

The system is now solved.

$$x=2,y=-\frac{21}{11}$$

Steps Using Matrices

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

$$4x+7\left(y+\frac{10}{11}\right)=1,5x-\left(y+\frac{10}{11}\right)=11$$

Simplify the first equation to put it in standard form.

$$4x+7\left(y+\frac{10}{11}\right)=1$$

Multiply 7 times y+\frac{10}{11}.

$$4x+7y+\frac{70}{11}=1$$

Subtract \frac{70}{11} from both sides of the equation.

$$4x+7y=-\frac{59}{11}$$

Simplify the second equation to put it in standard form.

$$5x-\left(y+\frac{10}{11}\right)=11$$

Multiply -1 times y+\frac{10}{11}.

$$5x-y-\frac{10}{11}=11$$

Add \frac{10}{11} to both sides of the equation.

$$5x-y=\frac{131}{11}$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&7\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&7\\5&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&7\\5&-1\end{matrix}\right))\left(\begin{matrix}4&7\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&-1\end{matrix}\right))\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&-1\end{matrix}\right))\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&7\\5&-1\end{matrix}\right))\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-7\times 5}&-\frac{7}{4\left(-1\right)-7\times 5}\\-\frac{5}{4\left(-1\right)-7\times 5}&\frac{4}{4\left(-1\right)-7\times 5}\end{matrix}\right)\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{39}&\frac{7}{39}\\\frac{5}{39}&-\frac{4}{39}\end{matrix}\right)\left(\begin{matrix}-\frac{59}{11}\\\frac{131}{11}\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{39}\left(-\frac{59}{11}\right)+\frac{7}{39}\times \frac{131}{11}\\\frac{5}{39}\left(-\frac{59}{11}\right)-\frac{4}{39}\times \frac{131}{11}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\-\frac{21}{11}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2,y=-\frac{21}{11}$$