Solve {l}{x+y=14.5}{70x+100y=2035}

$$\left. \begin{array} { l } { x + y = 14.5 } \\ { 70 x + 100 y = 2035 } \end{array} \right.$$

Solve for x, y

$$x=-19.5$$

$$y=34$$

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.

$$x+y=14.5,70x+100y=2035$$

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

$$x+y=14.5$$

Subtract y from both sides of the equation.

$$x=-y+14.5$$

Substitute -y+14.5 for x in the other equation, 70x+100y=2035.

$$70\left(-y+14.5\right)+100y=2035$$

Multiply 70 times -y+14.5.

$$-70y+1015+100y=2035$$

Add -70y to 100y.

$$30y+1015=2035$$

Subtract 1015 from both sides of the equation.

$$30y=1020$$

Divide both sides by 30.

$$y=34$$

Substitute 34 for y in x=-y+14.5. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-34+14.5$$

Add 14.5 to -34.

$$x=-19.5$$

The system is now solved.

$$x=-19.5,y=34$$

Steps Using Matrices

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

$$x+y=14.5,70x+100y=2035$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\70&100\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\70&100\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\70&100\end{matrix}\right))\left(\begin{matrix}1&1\\70&100\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\70&100\end{matrix}\right))\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\70&100\end{matrix}\right))\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\70&100\end{matrix}\right))\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{100}{100-70}&-\frac{1}{100-70}\\-\frac{70}{100-70}&\frac{1}{100-70}\end{matrix}\right)\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{3}&-\frac{1}{30}\\-\frac{7}{3}&\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}14.5\\2035\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{3}\times 14.5-\frac{1}{30}\times 2035\\-\frac{7}{3}\times 14.5+\frac{1}{30}\times 2035\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{39}{2}\\34\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{39}{2},y=34$$

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.

$$x+y=14.5,70x+100y=2035$$

To make x and 70x equal, multiply all terms on each side of the first equation by 70 and all terms on each side of the second by 1.

$$70x+70y=70\times 14.5,70x+100y=2035$$

Simplify.

$$70x+70y=1015,70x+100y=2035$$

Subtract 70x+100y=2035 from 70x+70y=1015 by subtracting like terms on each side of the equal sign.

$$70x-70x+70y-100y=1015-2035$$

Add 70x to -70x. Terms 70x and -70x cancel out, leaving an equation with only one variable that can be solved.

$$70y-100y=1015-2035$$

Add 70y to -100y.

$$-30y=1015-2035$$

Add 1015 to -2035.

$$-30y=-1020$$

Divide both sides by -30.

$$y=34$$

Substitute 34 for y in 70x+100y=2035. Because the resulting equation contains only one variable, you can solve for x directly.

$$70x+100\times 34=2035$$

Multiply 100 times 34.

$$70x+3400=2035$$

Subtract 3400 from both sides of the equation.

$$70x=-1365$$

Divide both sides by 70.

$$x=-\frac{39}{2}$$

The system is now solved.

$$x=-\frac{39}{2},y=34$$