Solve {l}{2x+4y=2060}{5x+7y=1640}

$$\left\{ \begin{array} { l } { 2 x + 4 y = 2060 } \\ { 5 x + 7 y = 1640 } \end{array} \right.$$

Solve for x, y

$$x=-1310$$

$$y=1170$$

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.

$$2x+4y=2060,5x+7y=1640$$

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

$$2x+4y=2060$$

Subtract 4y from both sides of the equation.

$$2x=-4y+2060$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-4y+2060\right)$$

Multiply \frac{1}{2} times -4y+2060.

$$x=-2y+1030$$

Substitute -2y+1030 for x in the other equation, 5x+7y=1640.

$$5\left(-2y+1030\right)+7y=1640$$

Multiply 5 times -2y+1030.

$$-10y+5150+7y=1640$$

Add -10y to 7y.

$$-3y+5150=1640$$

Subtract 5150 from both sides of the equation.

$$-3y=-3510$$

Divide both sides by -3.

$$y=1170$$

Substitute 1170 for y in x=-2y+1030. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-2\times 1170+1030$$

Multiply -2 times 1170.

$$x=-2340+1030$$

Add 1030 to -2340.

$$x=-1310$$

The system is now solved.

$$x=-1310,y=1170$$

Steps Using Matrices

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

$$2x+4y=2060,5x+7y=1640$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&4\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&4\\5&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&4\\5&7\end{matrix}\right))\left(\begin{matrix}2&4\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&7\end{matrix}\right))\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&7\end{matrix}\right))\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&4\\5&7\end{matrix}\right))\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-4\times 5}&-\frac{4}{2\times 7-4\times 5}\\-\frac{5}{2\times 7-4\times 5}&\frac{2}{2\times 7-4\times 5}\end{matrix}\right)\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{6}&\frac{2}{3}\\\frac{5}{6}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}2060\\1640\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{6}\times 2060+\frac{2}{3}\times 1640\\\frac{5}{6}\times 2060-\frac{1}{3}\times 1640\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1310\\1170\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-1310,y=1170$$

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.

$$2x+4y=2060,5x+7y=1640$$

To make 2x and 5x equal, multiply all terms on each side of the first equation by 5 and all terms on each side of the second by 2.

$$5\times 2x+5\times 4y=5\times 2060,2\times 5x+2\times 7y=2\times 1640$$

Simplify.

$$10x+20y=10300,10x+14y=3280$$

Subtract 10x+14y=3280 from 10x+20y=10300 by subtracting like terms on each side of the equal sign.

$$10x-10x+20y-14y=10300-3280$$

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

$$20y-14y=10300-3280$$

Add 20y to -14y.

$$6y=10300-3280$$

Add 10300 to -3280.

$$6y=7020$$

Divide both sides by 6.

$$y=1170$$

Substitute 1170 for y in 5x+7y=1640. Because the resulting equation contains only one variable, you can solve for x directly.

$$5x+7\times 1170=1640$$

Multiply 7 times 1170.

$$5x+8190=1640$$

Subtract 8190 from both sides of the equation.

$$5x=-6550$$

Divide both sides by 5.

$$x=-1310$$

The system is now solved.

$$x=-1310,y=1170$$