Solve {l}{-6u-14v=-12}{-8u+12v=76}

$$\left. \begin{array} { l } { - 6 u - 14 v = - 12 } \\ { - 8 u + 12 v = 76 } \end{array} \right.$$

Solve for u, v

$$u=-5$$

$$v=3$$

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.

$$-6u-14v=-12,-8u+12v=76$$

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

$$-6u-14v=-12$$

Add 14v to both sides of the equation.

$$-6u=14v-12$$

Divide both sides by -6.

$$u=-\frac{1}{6}\left(14v-12\right)$$

Multiply -\frac{1}{6} times 14v-12.

$$u=-\frac{7}{3}v+2$$

Substitute -\frac{7v}{3}+2 for u in the other equation, -8u+12v=76.

$$-8\left(-\frac{7}{3}v+2\right)+12v=76$$

Multiply -8 times -\frac{7v}{3}+2.

$$\frac{56}{3}v-16+12v=76$$

Add \frac{56v}{3} to 12v.

$$\frac{92}{3}v-16=76$$

Add 16 to both sides of the equation.

$$\frac{92}{3}v=92$$

Divide both sides of the equation by \frac{92}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$v=3$$

Substitute 3 for v in u=-\frac{7}{3}v+2. Because the resulting equation contains only one variable, you can solve for u directly.

$$u=-\frac{7}{3}\times 3+2$$

Multiply -\frac{7}{3} times 3.

$$u=-7+2$$

Add 2 to -7.

$$u=-5$$

The system is now solved.

$$u=-5,v=3$$

Steps Using Matrices

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

$$-6u-14v=-12,-8u+12v=76$$

Write the equations in matrix form.

$$\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-12\\76\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right))\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right))\left(\begin{matrix}-12\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right))\left(\begin{matrix}-12\\76\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\v\end{matrix}\right)=inverse(\left(\begin{matrix}-6&-14\\-8&12\end{matrix}\right))\left(\begin{matrix}-12\\76\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}\frac{12}{-6\times 12-\left(-14\left(-8\right)\right)}&-\frac{-14}{-6\times 12-\left(-14\left(-8\right)\right)}\\-\frac{-8}{-6\times 12-\left(-14\left(-8\right)\right)}&-\frac{6}{-6\times 12-\left(-14\left(-8\right)\right)}\end{matrix}\right)\left(\begin{matrix}-12\\76\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{46}&-\frac{7}{92}\\-\frac{1}{23}&\frac{3}{92}\end{matrix}\right)\left(\begin{matrix}-12\\76\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{46}\left(-12\right)-\frac{7}{92}\times 76\\-\frac{1}{23}\left(-12\right)+\frac{3}{92}\times 76\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}u\\v\end{matrix}\right)=\left(\begin{matrix}-5\\3\end{matrix}\right)$$

Extract the matrix elements u and v.

$$u=-5,v=3$$

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.

$$-6u-14v=-12,-8u+12v=76$$

To make -6u and -8u equal, multiply all terms on each side of the first equation by -8 and all terms on each side of the second by -6.

$$-8\left(-6\right)u-8\left(-14\right)v=-8\left(-12\right),-6\left(-8\right)u-6\times 12v=-6\times 76$$

Simplify.

$$48u+112v=96,48u-72v=-456$$

Subtract 48u-72v=-456 from 48u+112v=96 by subtracting like terms on each side of the equal sign.

$$48u-48u+112v+72v=96+456$$

Add 48u to -48u. Terms 48u and -48u cancel out, leaving an equation with only one variable that can be solved.

$$112v+72v=96+456$$

Add 112v to 72v.

$$184v=96+456$$

Add 96 to 456.

$$184v=552$$

Divide both sides by 184.

$$v=3$$

Substitute 3 for v in -8u+12v=76. Because the resulting equation contains only one variable, you can solve for u directly.

$$-8u+12\times 3=76$$

Multiply 12 times 3.

$$-8u+36=76$$

Subtract 36 from both sides of the equation.

$$-8u=40$$

Divide both sides by -8.

$$u=-5$$

The system is now solved.

$$u=-5,v=3$$