Solve 49x-57y=172;quad57x-49y=252

$$49 x - 57 y = 172 ; \quad 57 x - 49 y = 252$$

Solve for x, y

$$x=7$$

$$y=3$$

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.

$$49x-57y=172,57x-49y=252$$

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

$$49x-57y=172$$

Add 57y to both sides of the equation.

$$49x=57y+172$$

Divide both sides by 49.

$$x=\frac{1}{49}\left(57y+172\right)$$

Multiply \frac{1}{49} times 57y+172.

$$x=\frac{57}{49}y+\frac{172}{49}$$

Substitute \frac{57y+172}{49} for x in the other equation, 57x-49y=252.

$$57\left(\frac{57}{49}y+\frac{172}{49}\right)-49y=252$$

Multiply 57 times \frac{57y+172}{49}.

$$\frac{3249}{49}y+\frac{9804}{49}-49y=252$$

Add \frac{3249y}{49} to -49y.

$$\frac{848}{49}y+\frac{9804}{49}=252$$

Subtract \frac{9804}{49} from both sides of the equation.

$$\frac{848}{49}y=\frac{2544}{49}$$

Divide both sides of the equation by \frac{848}{49}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=3$$

Substitute 3 for y in x=\frac{57}{49}y+\frac{172}{49}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{57}{49}\times 3+\frac{172}{49}$$

Multiply \frac{57}{49} times 3.

$$x=\frac{171+172}{49}$$

Add \frac{172}{49} to \frac{171}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=7$$

The system is now solved.

$$x=7,y=3$$

Steps Using Matrices

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

$$49x-57y=172,57x-49y=252$$

Write the equations in matrix form.

$$\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}172\\252\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}49&-57\\57&-49\end{matrix}\right).

$$inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}49&-57\\57&-49\end{matrix}\right))\left(\begin{matrix}172\\252\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{49\left(-49\right)-\left(-57\times 57\right)}&-\frac{-57}{49\left(-49\right)-\left(-57\times 57\right)}\\-\frac{57}{49\left(-49\right)-\left(-57\times 57\right)}&\frac{49}{49\left(-49\right)-\left(-57\times 57\right)}\end{matrix}\right)\left(\begin{matrix}172\\252\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{848}&\frac{57}{848}\\-\frac{57}{848}&\frac{49}{848}\end{matrix}\right)\left(\begin{matrix}172\\252\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{49}{848}\times 172+\frac{57}{848}\times 252\\-\frac{57}{848}\times 172+\frac{49}{848}\times 252\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=7,y=3$$

Steps Using Elimination

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.

$$49x-57y=172,57x-49y=252$$

To make 49x and 57x equal, multiply all terms on each side of the first equation by 57 and all terms on each side of the second by 49.

$$57\times 49x+57\left(-57\right)y=57\times 172,49\times 57x+49\left(-49\right)y=49\times 252$$

Simplify.

$$2793x-3249y=9804,2793x-2401y=12348$$

Subtract 2793x-2401y=12348 from 2793x-3249y=9804 by subtracting like terms on each side of the equal sign.

$$2793x-2793x-3249y+2401y=9804-12348$$

Add 2793x to -2793x. Terms 2793x and -2793x cancel out, leaving an equation with only one variable that can be solved.

$$-3249y+2401y=9804-12348$$

Add -3249y to 2401y.

$$-848y=9804-12348$$

Add 9804 to -12348.

$$-848y=-2544$$

Divide both sides by -848.

$$y=3$$

Substitute 3 for y in 57x-49y=252. Because the resulting equation contains only one variable, you can solve for x directly.

$$57x-49\times 3=252$$

Multiply -49 times 3.

$$57x-147=252$$

Add 147 to both sides of the equation.

$$57x=399$$

Divide both sides by 57.

$$x=7$$

The system is now solved.

$$x=7,y=3$$