Solve {l}{2.5x+12.5y=9750}{13x-13y=9750}

$$\left\{ \begin{array} { l } { 2.5 x + 12.5 y = 9750 } \\ { 13 x - 13 y = 9750 } \end{array} \right.$$

Solve for x, y

$$x=1275$$

$$y=525$$

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.

$$2.5x+12.5y=9750,13x-13y=9750$$

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

$$2.5x+12.5y=9750$$

Subtract \frac{25y}{2} from both sides of the equation.

$$2.5x=-12.5y+9750$$

Divide both sides of the equation by 2.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=0.4\left(-12.5y+9750\right)$$

Multiply 0.4 times -\frac{25y}{2}+9750.

$$x=-5y+3900$$

Substitute -5y+3900 for x in the other equation, 13x-13y=9750.

$$13\left(-5y+3900\right)-13y=9750$$

Multiply 13 times -5y+3900.

$$-65y+50700-13y=9750$$

Add -65y to -13y.

$$-78y+50700=9750$$

Subtract 50700 from both sides of the equation.

$$-78y=-40950$$

Divide both sides by -78.

$$y=525$$

Substitute 525 for y in x=-5y+3900. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-5\times 525+3900$$

Multiply -5 times 525.

$$x=-2625+3900$$

Add 3900 to -2625.

$$x=1275$$

The system is now solved.

$$x=1275,y=525$$

Steps Using Matrices

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

$$2.5x+12.5y=9750,13x-13y=9750$$

Write the equations in matrix form.

$$\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right).

$$inverse(\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right))\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right))\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right))\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2.5&12.5\\13&-13\end{matrix}\right))\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{2.5\left(-13\right)-12.5\times 13}&-\frac{12.5}{2.5\left(-13\right)-12.5\times 13}\\-\frac{13}{2.5\left(-13\right)-12.5\times 13}&\frac{2.5}{2.5\left(-13\right)-12.5\times 13}\end{matrix}\right)\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}&\frac{5}{78}\\\frac{1}{15}&-\frac{1}{78}\end{matrix}\right)\left(\begin{matrix}9750\\9750\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{15}\times 9750+\frac{5}{78}\times 9750\\\frac{1}{15}\times 9750-\frac{1}{78}\times 9750\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1275\\525\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1275,y=525$$

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.

$$2.5x+12.5y=9750,13x-13y=9750$$

To make \frac{5x}{2} and 13x equal, multiply all terms on each side of the first equation by 13 and all terms on each side of the second by 2.5.

$$13\times 2.5x+13\times 12.5y=13\times 9750,2.5\times 13x+2.5\left(-13\right)y=2.5\times 9750$$

Simplify.

$$32.5x+162.5y=126750,32.5x-32.5y=24375$$

Subtract 32.5x-32.5y=24375 from 32.5x+162.5y=126750 by subtracting like terms on each side of the equal sign.

$$32.5x-32.5x+162.5y+32.5y=126750-24375$$

Add \frac{65x}{2} to -\frac{65x}{2}. Terms \frac{65x}{2} and -\frac{65x}{2} cancel out, leaving an equation with only one variable that can be solved.

$$162.5y+32.5y=126750-24375$$

Add \frac{325y}{2} to \frac{65y}{2}.

$$195y=126750-24375$$

Add 126750 to -24375.

$$195y=102375$$

Divide both sides by 195.

$$y=525$$

Substitute 525 for y in 13x-13y=9750. Because the resulting equation contains only one variable, you can solve for x directly.

$$13x-13\times 525=9750$$

Multiply -13 times 525.

$$13x-6825=9750$$

Add 6825 to both sides of the equation.

$$13x=16575$$

Divide both sides by 13.

$$x=1275$$

The system is now solved.

$$x=1275,y=525$$