Solve {l}{5x+7y=2060}{2060-2x-4y=120}

$$\left\{ \begin{array} { l } { 5 x + 7 y = 2060 } \\ { 2060 - 2 x - 4 y = 120 } \end{array} \right.$$

Solve for x, y

$$x=-890$$

$$y=930$$

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.

$$5x+7y=2060,-2x-4y+2060=120$$

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

$$5x+7y=2060$$

Subtract 7y from both sides of the equation.

$$5x=-7y+2060$$

Divide both sides by 5.

$$x=\frac{1}{5}\left(-7y+2060\right)$$

Multiply \frac{1}{5} times -7y+2060.

$$x=-\frac{7}{5}y+412$$

Substitute -\frac{7y}{5}+412 for x in the other equation, -2x-4y+2060=120.

$$-2\left(-\frac{7}{5}y+412\right)-4y+2060=120$$

Multiply -2 times -\frac{7y}{5}+412.

$$\frac{14}{5}y-824-4y+2060=120$$

Add \frac{14y}{5} to -4y.

$$-\frac{6}{5}y-824+2060=120$$

Add -824 to 2060.

$$-\frac{6}{5}y+1236=120$$

Subtract 1236 from both sides of the equation.

$$-\frac{6}{5}y=-1116$$

Divide both sides of the equation by -\frac{6}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=930$$

Substitute 930 for y in x=-\frac{7}{5}y+412. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{7}{5}\times 930+412$$

Multiply -\frac{7}{5} times 930.

$$x=-1302+412$$

Add 412 to -1302.

$$x=-890$$

The system is now solved.

$$x=-890,y=930$$

Steps Using Matrices

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

$$5x+7y=2060,-2x-4y+2060=120$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&7\\-2&-4\end{matrix}\right))\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{5\left(-4\right)-7\left(-2\right)}&-\frac{7}{5\left(-4\right)-7\left(-2\right)}\\-\frac{-2}{5\left(-4\right)-7\left(-2\right)}&\frac{5}{5\left(-4\right)-7\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}&\frac{7}{6}\\-\frac{1}{3}&-\frac{5}{6}\end{matrix}\right)\left(\begin{matrix}2060\\-1940\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3}\times 2060+\frac{7}{6}\left(-1940\right)\\-\frac{1}{3}\times 2060-\frac{5}{6}\left(-1940\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-890\\930\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-890,y=930$$

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.

$$5x+7y=2060,-2x-4y+2060=120$$

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

$$-2\times 5x-2\times 7y=-2\times 2060,5\left(-2\right)x+5\left(-4\right)y+5\times 2060=5\times 120$$

Simplify.

$$-10x-14y=-4120,-10x-20y+10300=600$$

Subtract -10x-20y+10300=600 from -10x-14y=-4120 by subtracting like terms on each side of the equal sign.

$$-10x+10x-14y+20y-10300=-4120-600$$

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

$$-14y+20y-10300=-4120-600$$

Add -14y to 20y.

$$6y-10300=-4120-600$$

Add -4120 to -600.

$$6y-10300=-4720$$

Add 10300 to both sides of the equation.

$$6y=5580$$

Divide both sides by 6.

$$y=930$$

Substitute 930 for y in -2x-4y+2060=120. Because the resulting equation contains only one variable, you can solve for x directly.

$$-2x-4\times 930+2060=120$$

Multiply -4 times 930.

$$-2x-3720+2060=120$$

Add -3720 to 2060.

$$-2x-1660=120$$

Add 1660 to both sides of the equation.

$$-2x=1780$$

Divide both sides by -2.

$$x=-890$$

The system is now solved.

$$x=-890,y=930$$