Solve {l}{10x+15y=2050}{x+y=175}

$$\left. \begin{array}{l}{ 10 x + 15 y = 2050 }\\{ x + y = 175 }\end{array} \right.$$

Solve for x, y

$$x=115$$

$$y=60$$

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.

$$10x+15y=2050,x+y=175$$

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

$$10x+15y=2050$$

Subtract 15y from both sides of the equation.

$$10x=-15y+2050$$

Divide both sides by 10.

$$x=\frac{1}{10}\left(-15y+2050\right)$$

Multiply \frac{1}{10} times -15y+2050.

$$x=-\frac{3}{2}y+205$$

Substitute -\frac{3y}{2}+205 for x in the other equation, x+y=175.

$$-\frac{3}{2}y+205+y=175$$

Add -\frac{3y}{2} to y.

$$-\frac{1}{2}y+205=175$$

Subtract 205 from both sides of the equation.

$$-\frac{1}{2}y=-30$$

Multiply both sides by -2.

$$y=60$$

Substitute 60 for y in x=-\frac{3}{2}y+205. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times 60+205$$

Multiply -\frac{3}{2} times 60.

$$x=-90+205$$

Add 205 to -90.

$$x=115$$

The system is now solved.

$$x=115,y=60$$

Steps Using Matrices

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

$$10x+15y=2050,x+y=175$$

Write the equations in matrix form.

$$\left(\begin{matrix}10&15\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2050\\175\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&15\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&15\\1&1\end{matrix}\right))\left(\begin{matrix}10&15\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&15\\1&1\end{matrix}\right))\left(\begin{matrix}2050\\175\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&15\\1&1\end{matrix}\right))\left(\begin{matrix}2050\\175\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&15\\1&1\end{matrix}\right))\left(\begin{matrix}2050\\175\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{10-15}&-\frac{15}{10-15}\\-\frac{1}{10-15}&\frac{10}{10-15}\end{matrix}\right)\left(\begin{matrix}2050\\175\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&3\\\frac{1}{5}&-2\end{matrix}\right)\left(\begin{matrix}2050\\175\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 2050+3\times 175\\\frac{1}{5}\times 2050-2\times 175\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}115\\60\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=115,y=60$$

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.

$$10x+15y=2050,x+y=175$$

To make 10x and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 10.

$$10x+15y=2050,10x+10y=10\times 175$$

Simplify.

$$10x+15y=2050,10x+10y=1750$$

Subtract 10x+10y=1750 from 10x+15y=2050 by subtracting like terms on each side of the equal sign.

$$10x-10x+15y-10y=2050-1750$$

Add 10x to -10x. Terms 10x and -10x cancel out, leaving an equation with only one variable that can be solved.

$$15y-10y=2050-1750$$

Add 15y to -10y.

$$5y=2050-1750$$

Add 2050 to -1750.

$$5y=300$$

Divide both sides by 5.

$$y=60$$

Substitute 60 for y in x+y=175. Because the resulting equation contains only one variable, you can solve for x directly.

$$x+60=175$$

Subtract 60 from both sides of the equation.

$$x=115$$

The system is now solved.

$$x=115,y=60$$