Solve {l}{173x+32y=148}{216x+642y=426}

$$\left\{ \begin{array} { l } { 173 x + 32 y = 148 } \\ { 216 x + 642 y = 426 } \end{array} \right.$$

Solve for x, y

$$x=\frac{13564}{17359}\approx 0.781381416$$

$$y=\frac{6955}{17359}\approx 0.40065672$$

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.

$$173x+32y=148,216x+642y=426$$

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

$$173x+32y=148$$

Subtract 32y from both sides of the equation.

$$173x=-32y+148$$

Divide both sides by 173.

$$x=\frac{1}{173}\left(-32y+148\right)$$

Multiply \frac{1}{173} times -32y+148.

$$x=-\frac{32}{173}y+\frac{148}{173}$$

Substitute \frac{-32y+148}{173} for x in the other equation, 216x+642y=426.

$$216\left(-\frac{32}{173}y+\frac{148}{173}\right)+642y=426$$

Multiply 216 times \frac{-32y+148}{173}.

$$-\frac{6912}{173}y+\frac{31968}{173}+642y=426$$

Add -\frac{6912y}{173} to 642y.

$$\frac{104154}{173}y+\frac{31968}{173}=426$$

Subtract \frac{31968}{173} from both sides of the equation.

$$\frac{104154}{173}y=\frac{41730}{173}$$

Divide both sides of the equation by \frac{104154}{173}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{6955}{17359}$$

Substitute \frac{6955}{17359} for y in x=-\frac{32}{173}y+\frac{148}{173}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{32}{173}\times \frac{6955}{17359}+\frac{148}{173}$$

Multiply -\frac{32}{173} times \frac{6955}{17359} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{222560}{3003107}+\frac{148}{173}$$

Add \frac{148}{173} to -\frac{222560}{3003107} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{13564}{17359}$$

The system is now solved.

$$x=\frac{13564}{17359},y=\frac{6955}{17359}$$

Steps Using Matrices

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

$$173x+32y=148,216x+642y=426$$

Write the equations in matrix form.

$$\left(\begin{matrix}173&32\\216&642\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}148\\426\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}173&32\\216&642\end{matrix}\right).

$$inverse(\left(\begin{matrix}173&32\\216&642\end{matrix}\right))\left(\begin{matrix}173&32\\216&642\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}173&32\\216&642\end{matrix}\right))\left(\begin{matrix}148\\426\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}173&32\\216&642\end{matrix}\right))\left(\begin{matrix}148\\426\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}173&32\\216&642\end{matrix}\right))\left(\begin{matrix}148\\426\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{642}{173\times 642-32\times 216}&-\frac{32}{173\times 642-32\times 216}\\-\frac{216}{173\times 642-32\times 216}&\frac{173}{173\times 642-32\times 216}\end{matrix}\right)\left(\begin{matrix}148\\426\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{107}{17359}&-\frac{16}{52077}\\-\frac{36}{17359}&\frac{173}{104154}\end{matrix}\right)\left(\begin{matrix}148\\426\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{107}{17359}\times 148-\frac{16}{52077}\times 426\\-\frac{36}{17359}\times 148+\frac{173}{104154}\times 426\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13564}{17359}\\\frac{6955}{17359}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{13564}{17359},y=\frac{6955}{17359}$$

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.

$$173x+32y=148,216x+642y=426$$

To make 173x and 216x equal, multiply all terms on each side of the first equation by 216 and all terms on each side of the second by 173.

$$216\times 173x+216\times 32y=216\times 148,173\times 216x+173\times 642y=173\times 426$$

Simplify.

$$37368x+6912y=31968,37368x+111066y=73698$$

Subtract 37368x+111066y=73698 from 37368x+6912y=31968 by subtracting like terms on each side of the equal sign.

$$37368x-37368x+6912y-111066y=31968-73698$$

Add 37368x to -37368x. Terms 37368x and -37368x cancel out, leaving an equation with only one variable that can be solved.

$$6912y-111066y=31968-73698$$

Add 6912y to -111066y.

$$-104154y=31968-73698$$

Add 31968 to -73698.

$$-104154y=-41730$$

Divide both sides by -104154.

$$y=\frac{6955}{17359}$$

Substitute \frac{6955}{17359} for y in 216x+642y=426. Because the resulting equation contains only one variable, you can solve for x directly.

$$216x+642\times \frac{6955}{17359}=426$$

Multiply 642 times \frac{6955}{17359}.

$$216x+\frac{4465110}{17359}=426$$

Subtract \frac{4465110}{17359} from both sides of the equation.

$$216x=\frac{2929824}{17359}$$

Divide both sides by 216.

$$x=\frac{13564}{17359}$$

The system is now solved.

$$x=\frac{13564}{17359},y=\frac{6955}{17359}$$