Solve {l}{x/33-y=26}{x-y/88=66}

$$\left. \begin{array} { l } { \frac { x } { 33 } - y = 26 } \\ { x - \frac { y } { 88 } = 66 } \end{array} \right.$$

Solve for x, y

$$x = \frac{190806}{2903} = 65\frac{2111}{2903} \approx 65.727178781$$

$$y = -\frac{69696}{2903} = -24\frac{24}{2903} \approx -24.00826731$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 33.

$$x-33y=858$$

Consider the second equation. Multiply both sides of the equation by 88.

$$88x-y=5808$$

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.

$$x-33y=858,88x-y=5808$$

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

$$x-33y=858$$

Add 33y to both sides of the equation.

$$x=33y+858$$

Substitute 858+33y for x in the other equation, 88x-y=5808.

$$88\left(33y+858\right)-y=5808$$

Multiply 88 times 858+33y.

$$2904y+75504-y=5808$$

Add 2904y to -y.

$$2903y+75504=5808$$

Subtract 75504 from both sides of the equation.

$$2903y=-69696$$

Divide both sides by 2903.

$$y=-\frac{69696}{2903}$$

Substitute -\frac{69696}{2903} for y in x=33y+858. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=33\left(-\frac{69696}{2903}\right)+858$$

Multiply 33 times -\frac{69696}{2903}.

$$x=-\frac{2299968}{2903}+858$$

Add 858 to -\frac{2299968}{2903}.

$$x=\frac{190806}{2903}$$

The system is now solved.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 33.

$$x-33y=858$$

Consider the second equation. Multiply both sides of the equation by 88.

$$88x-y=5808$$

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

$$x-33y=858,88x-y=5808$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-33\\88&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-33\\88&-1\end{matrix}\right))\left(\begin{matrix}858\\5808\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-33\times 88\right)}&-\frac{-33}{-1-\left(-33\times 88\right)}\\-\frac{88}{-1-\left(-33\times 88\right)}&\frac{1}{-1-\left(-33\times 88\right)}\end{matrix}\right)\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2903}&\frac{33}{2903}\\-\frac{88}{2903}&\frac{1}{2903}\end{matrix}\right)\left(\begin{matrix}858\\5808\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2903}\times 858+\frac{33}{2903}\times 5808\\-\frac{88}{2903}\times 858+\frac{1}{2903}\times 5808\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{190806}{2903}\\-\frac{69696}{2903}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 33.

$$x-33y=858$$

Consider the second equation. Multiply both sides of the equation by 88.

$$88x-y=5808$$

In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.

$$x-33y=858,88x-y=5808$$

To make x and 88x equal, multiply all terms on each side of the first equation by 88 and all terms on each side of the second by 1.

$$88x+88\left(-33\right)y=88\times 858,88x-y=5808$$

Simplify.

$$88x-2904y=75504,88x-y=5808$$

Subtract 88x-y=5808 from 88x-2904y=75504 by subtracting like terms on each side of the equal sign.

$$88x-88x-2904y+y=75504-5808$$

Add 88x to -88x. Terms 88x and -88x cancel out, leaving an equation with only one variable that can be solved.

$$-2904y+y=75504-5808$$

Add -2904y to y.

$$-2903y=75504-5808$$

Add 75504 to -5808.

$$-2903y=69696$$

Divide both sides by -2903.

$$y=-\frac{69696}{2903}$$

Substitute -\frac{69696}{2903} for y in 88x-y=5808. Because the resulting equation contains only one variable, you can solve for x directly.

$$88x-\left(-\frac{69696}{2903}\right)=5808$$

Subtract \frac{69696}{2903} from both sides of the equation.

$$88x=\frac{16790928}{2903}$$

Divide both sides by 88.

$$x=\frac{190806}{2903}$$

The system is now solved.

$$x=\frac{190806}{2903},y=-\frac{69696}{2903}$$