Solve {l}{82x+49y=36}{48x-26y=102}

$$\left. \begin{array} { l } { 82 x + 49 y = 36 } \\ { 48 x - 26 y = 102 } \end{array} \right.$$

Solve for x, y

$$x = \frac{2967}{2242} = 1\frac{725}{2242} \approx 1.323371989$$

$$y = -\frac{1659}{1121} = -1\frac{538}{1121} \approx -1.479928635$$

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.

$$82x+49y=36,48x-26y=102$$

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

$$82x+49y=36$$

Subtract 49y from both sides of the equation.

$$82x=-49y+36$$

Divide both sides by 82.

$$x=\frac{1}{82}\left(-49y+36\right)$$

Multiply \frac{1}{82} times -49y+36.

$$x=-\frac{49}{82}y+\frac{18}{41}$$

Substitute -\frac{49y}{82}+\frac{18}{41} for x in the other equation, 48x-26y=102.

$$48\left(-\frac{49}{82}y+\frac{18}{41}\right)-26y=102$$

Multiply 48 times -\frac{49y}{82}+\frac{18}{41}.

$$-\frac{1176}{41}y+\frac{864}{41}-26y=102$$

Add -\frac{1176y}{41} to -26y.

$$-\frac{2242}{41}y+\frac{864}{41}=102$$

Subtract \frac{864}{41} from both sides of the equation.

$$-\frac{2242}{41}y=\frac{3318}{41}$$

Divide both sides of the equation by -\frac{2242}{41}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{1659}{1121}$$

Substitute -\frac{1659}{1121} for y in x=-\frac{49}{82}y+\frac{18}{41}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{49}{82}\left(-\frac{1659}{1121}\right)+\frac{18}{41}$$

Multiply -\frac{49}{82} times -\frac{1659}{1121} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{81291}{91922}+\frac{18}{41}$$

Add \frac{18}{41} to \frac{81291}{91922} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{2967}{2242}$$

The system is now solved.

$$x=\frac{2967}{2242},y=-\frac{1659}{1121}$$

Steps Using Matrices

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

$$82x+49y=36,48x-26y=102$$

Write the equations in matrix form.

$$\left(\begin{matrix}82&49\\48&-26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}36\\102\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}82&49\\48&-26\end{matrix}\right).

$$inverse(\left(\begin{matrix}82&49\\48&-26\end{matrix}\right))\left(\begin{matrix}82&49\\48&-26\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}82&49\\48&-26\end{matrix}\right))\left(\begin{matrix}36\\102\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}82&49\\48&-26\end{matrix}\right))\left(\begin{matrix}36\\102\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}82&49\\48&-26\end{matrix}\right))\left(\begin{matrix}36\\102\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{26}{82\left(-26\right)-49\times 48}&-\frac{49}{82\left(-26\right)-49\times 48}\\-\frac{48}{82\left(-26\right)-49\times 48}&\frac{82}{82\left(-26\right)-49\times 48}\end{matrix}\right)\left(\begin{matrix}36\\102\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{2242}&\frac{49}{4484}\\\frac{12}{1121}&-\frac{41}{2242}\end{matrix}\right)\left(\begin{matrix}36\\102\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{2242}\times 36+\frac{49}{4484}\times 102\\\frac{12}{1121}\times 36-\frac{41}{2242}\times 102\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2967}{2242}\\-\frac{1659}{1121}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{2967}{2242},y=-\frac{1659}{1121}$$

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.

$$82x+49y=36,48x-26y=102$$

To make 82x and 48x equal, multiply all terms on each side of the first equation by 48 and all terms on each side of the second by 82.

$$48\times 82x+48\times 49y=48\times 36,82\times 48x+82\left(-26\right)y=82\times 102$$

Simplify.

$$3936x+2352y=1728,3936x-2132y=8364$$

Subtract 3936x-2132y=8364 from 3936x+2352y=1728 by subtracting like terms on each side of the equal sign.

$$3936x-3936x+2352y+2132y=1728-8364$$

Add 3936x to -3936x. Terms 3936x and -3936x cancel out, leaving an equation with only one variable that can be solved.

$$2352y+2132y=1728-8364$$

Add 2352y to 2132y.

$$4484y=1728-8364$$

Add 1728 to -8364.

$$4484y=-6636$$

Divide both sides by 4484.

$$y=-\frac{1659}{1121}$$

Substitute -\frac{1659}{1121} for y in 48x-26y=102. Because the resulting equation contains only one variable, you can solve for x directly.

$$48x-26\left(-\frac{1659}{1121}\right)=102$$

Multiply -26 times -\frac{1659}{1121}.

$$48x+\frac{43134}{1121}=102$$

Subtract \frac{43134}{1121} from both sides of the equation.

$$48x=\frac{71208}{1121}$$

Divide both sides by 48.

$$x=\frac{2967}{2242}$$

The system is now solved.

$$x=\frac{2967}{2242},y=-\frac{1659}{1121}$$