Solve {l}{4x-6y=40}{6x-2y=140}

$$\left. \begin{array} { l } { 4 x - 6 y = 40 } \\ { 6 x - 2 y = 140 } \end{array} \right.$$

Solve for x, y

$$x = \frac{190}{7} = 27\frac{1}{7} \approx 27.142857143$$

$$y = \frac{80}{7} = 11\frac{3}{7} \approx 11.428571429$$

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.

$$4x-6y=40,6x-2y=140$$

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

$$4x-6y=40$$

Add 6y to both sides of the equation.

$$4x=6y+40$$

Divide both sides by 4.

$$x=\frac{1}{4}\left(6y+40\right)$$

Multiply \frac{1}{4} times 6y+40.

$$x=\frac{3}{2}y+10$$

Substitute \frac{3y}{2}+10 for x in the other equation, 6x-2y=140.

$$6\left(\frac{3}{2}y+10\right)-2y=140$$

Multiply 6 times \frac{3y}{2}+10.

$$9y+60-2y=140$$

Add 9y to -2y.

$$7y+60=140$$

Subtract 60 from both sides of the equation.

$$7y=80$$

Divide both sides by 7.

$$y=\frac{80}{7}$$

Substitute \frac{80}{7} for y in x=\frac{3}{2}y+10. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{3}{2}\times \frac{80}{7}+10$$

Multiply \frac{3}{2} times \frac{80}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{120}{7}+10$$

Add 10 to \frac{120}{7}.

$$x=\frac{190}{7}$$

The system is now solved.

$$x=\frac{190}{7},y=\frac{80}{7}$$

Steps Using Matrices

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

$$4x-6y=40,6x-2y=140$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\140\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}4&-6\\6&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right))\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right))\left(\begin{matrix}40\\140\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right))\left(\begin{matrix}40\\140\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-6\\6&-2\end{matrix}\right))\left(\begin{matrix}40\\140\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{4\left(-2\right)-\left(-6\times 6\right)}&-\frac{-6}{4\left(-2\right)-\left(-6\times 6\right)}\\-\frac{6}{4\left(-2\right)-\left(-6\times 6\right)}&\frac{4}{4\left(-2\right)-\left(-6\times 6\right)}\end{matrix}\right)\left(\begin{matrix}40\\140\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}&\frac{3}{14}\\-\frac{3}{14}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}40\\140\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{14}\times 40+\frac{3}{14}\times 140\\-\frac{3}{14}\times 40+\frac{1}{7}\times 140\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{190}{7}\\\frac{80}{7}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{190}{7},y=\frac{80}{7}$$

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.

$$4x-6y=40,6x-2y=140$$

To make 4x and 6x equal, multiply all terms on each side of the first equation by 6 and all terms on each side of the second by 4.

$$6\times 4x+6\left(-6\right)y=6\times 40,4\times 6x+4\left(-2\right)y=4\times 140$$

Simplify.

$$24x-36y=240,24x-8y=560$$

Subtract 24x-8y=560 from 24x-36y=240 by subtracting like terms on each side of the equal sign.

$$24x-24x-36y+8y=240-560$$

Add 24x to -24x. Terms 24x and -24x cancel out, leaving an equation with only one variable that can be solved.

$$-36y+8y=240-560$$

Add -36y to 8y.

$$-28y=240-560$$

Add 240 to -560.

$$-28y=-320$$

Divide both sides by -28.

$$y=\frac{80}{7}$$

Substitute \frac{80}{7} for y in 6x-2y=140. Because the resulting equation contains only one variable, you can solve for x directly.

$$6x-2\times \frac{80}{7}=140$$

Multiply -2 times \frac{80}{7}.

$$6x-\frac{160}{7}=140$$

Add \frac{160}{7} to both sides of the equation.

$$6x=\frac{1140}{7}$$

Divide both sides by 6.

$$x=\frac{190}{7}$$

The system is now solved.

$$x=\frac{190}{7},y=\frac{80}{7}$$