Solve {l}{64x+2py=1.52}{98x+68y=2154}

$$\left\{ \begin{array} { l } { 64 x + 2 p y = 1.52 } \\ { 98 x + 68 y = 2154 } \end{array} \right.$$

Solve for x, y

$$x=-\frac{646-26925p}{25\left(49p-1088\right)}$$

$$y=-\frac{860669}{25\left(49p-1088\right)}$$

$$p\neq \frac{1088}{49}$$

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.

$$64x+2py=1.52,98x+68y=2154$$

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

$$64x+2py=1.52$$

Subtract 2py from both sides of the equation.

$$64x=\left(-2p\right)y+1.52$$

Divide both sides by 64.

$$x=\frac{1}{64}\left(\left(-2p\right)y+1.52\right)$$

Multiply \frac{1}{64} times -2py+1.52.

$$x=\left(-\frac{p}{32}\right)y+\frac{19}{800}$$

Substitute -\frac{py}{32}+\frac{19}{800} for x in the other equation, 98x+68y=2154.

$$98\left(\left(-\frac{p}{32}\right)y+\frac{19}{800}\right)+68y=2154$$

Multiply 98 times -\frac{py}{32}+\frac{19}{800}.

$$\left(-\frac{49p}{16}\right)y+\frac{931}{400}+68y=2154$$

Add -\frac{49py}{16} to 68y.

$$\left(-\frac{49p}{16}+68\right)y+\frac{931}{400}=2154$$

Subtract \frac{931}{400} from both sides of the equation.

$$\left(-\frac{49p}{16}+68\right)y=\frac{860669}{400}$$

Divide both sides by -\frac{49p}{16}+68.

$$y=\frac{860669}{25\left(1088-49p\right)}$$

Substitute \frac{860669}{25\left(1088-49p\right)} for y in x=\left(-\frac{p}{32}\right)y+\frac{19}{800}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-\frac{p}{32}\right)\times \frac{860669}{25\left(1088-49p\right)}+\frac{19}{800}$$

Multiply -\frac{p}{32} times \frac{860669}{25\left(1088-49p\right)}.

$$x=-\frac{860669p}{800\left(1088-49p\right)}+\frac{19}{800}$$

Add \frac{19}{800} to -\frac{860669p}{800\left(1088-49p\right)}.

$$x=\frac{646-26925p}{25\left(1088-49p\right)}$$

The system is now solved.

$$x=\frac{646-26925p}{25\left(1088-49p\right)},y=\frac{860669}{25\left(1088-49p\right)}$$

Steps Using Matrices

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

$$64x+2py=1.52,98x+68y=2154$$

Write the equations in matrix form.

$$\left(\begin{matrix}64&2p\\98&68\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}64&2p\\98&68\end{matrix}\right).

$$inverse(\left(\begin{matrix}64&2p\\98&68\end{matrix}\right))\left(\begin{matrix}64&2p\\98&68\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&2p\\98&68\end{matrix}\right))\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&2p\\98&68\end{matrix}\right))\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}64&2p\\98&68\end{matrix}\right))\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{68}{64\times 68-2p\times 98}&-\frac{2p}{64\times 68-2p\times 98}\\-\frac{98}{64\times 68-2p\times 98}&\frac{64}{64\times 68-2p\times 98}\end{matrix}\right)\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{1088-49p}&-\frac{p}{2\left(1088-49p\right)}\\-\frac{49}{2\left(1088-49p\right)}&\frac{16}{1088-49p}\end{matrix}\right)\left(\begin{matrix}1.52\\2154\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{1088-49p}\times 1.52+\left(-\frac{p}{2\left(1088-49p\right)}\right)\times 2154\\\left(-\frac{49}{2\left(1088-49p\right)}\right)\times 1.52+\frac{16}{1088-49p}\times 2154\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{211092p-5064.64}{196\left(1088-49p\right)}\\\frac{860669}{25\left(1088-49p\right)}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{211092p-5064.64}{196\left(49p-1088\right)},y=\frac{860669}{25\left(1088-49p\right)}$$

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.

$$64x+2py=1.52,98x+68y=2154$$

To make 64x and 98x equal, multiply all terms on each side of the first equation by 98 and all terms on each side of the second by 64.

$$98\times 64x+98\times 2py=98\times 1.52,64\times 98x+64\times 68y=64\times 2154$$

Simplify.

$$6272x+196py=148.96,6272x+4352y=137856$$

Subtract 6272x+4352y=137856 from 6272x+196py=148.96 by subtracting like terms on each side of the equal sign.

$$6272x-6272x+196py-4352y=148.96-137856$$

Add 6272x to -6272x. Terms 6272x and -6272x cancel out, leaving an equation with only one variable that can be solved.

$$196py-4352y=148.96-137856$$

Add 196py to -4352y.

$$\left(196p-4352\right)y=148.96-137856$$

Add 148.96 to -137856.

$$\left(196p-4352\right)y=-137707.04$$

Divide both sides by 196p-4352.

$$y=-\frac{860669}{25\left(49p-1088\right)}$$

Substitute -\frac{860669}{25\left(-1088+49p\right)} for y in 98x+68y=2154. Because the resulting equation contains only one variable, you can solve for x directly.

$$98x+68\left(-\frac{860669}{25\left(49p-1088\right)}\right)=2154$$

Multiply 68 times -\frac{860669}{25\left(-1088+49p\right)}.

$$98x-\frac{58525492}{25\left(49p-1088\right)}=2154$$

Add \frac{58525492}{25\left(-1088+49p\right)} to both sides of the equation.

$$98x=\frac{98\left(26925p-646\right)}{25\left(49p-1088\right)}$$

Divide both sides by 98.

$$x=\frac{26925p-646}{25\left(49p-1088\right)}$$

The system is now solved.

$$x=\frac{26925p-646}{25\left(49p-1088\right)},y=-\frac{860669}{25\left(49p-1088\right)}$$