Solve {l}{6-2L+4lambda=0}{20-4K+8lambda=0}{42+8K-88=0}

$$\left\{ \begin{array} { l } { 6 - 2 L + 4 \lambda = 0 } \\ { 20 - 4 K + 8 \lambda = 0 } \\ { 42 + 8 K - 88 = 0 } \end{array} \right.$$

Solve for L, λ, K

$$L = \frac{15}{4} = 3\frac{3}{4} = 3.75$$

$$\lambda =\frac{3}{8}=0.375$$

$$K = \frac{23}{4} = 5\frac{3}{4} = 5.75$$

Solution Steps

Consider the third equation. Subtract 88 from 42 to get -46.

$$-46+8K=0$$

Add 46 to both sides. Anything plus zero gives itself.

$$8K=46$$

Divide both sides by 8.

$$K=\frac{46}{8}$$

Reduce the fraction \frac{46}{8} to lowest terms by extracting and canceling out 2.

$$K=\frac{23}{4}$$

Consider the second equation. Insert the known values of variables into the equation.

$$20-4\times \frac{23}{4}+8\lambda =0$$

Multiply -4 and \frac{23}{4} to get -23.

$$20-23+8\lambda =0$$

Subtract 23 from 20 to get -3.

$$-3+8\lambda =0$$

Add 3 to both sides. Anything plus zero gives itself.

$$8\lambda =3$$

Divide both sides by 8.

$$\lambda =\frac{3}{8}$$

Consider the first equation. Insert the known values of variables into the equation.

$$6-2L+4\times \frac{3}{8}=0$$

Multiply 4 and \frac{3}{8} to get \frac{3}{2}.

$$6-2L+\frac{3}{2}=0$$

Add 6 and \frac{3}{2} to get \frac{15}{2}.

$$\frac{15}{2}-2L=0$$

Subtract \frac{15}{2} from both sides. Anything subtracted from zero gives its negation.

$$-2L=-\frac{15}{2}$$

Divide both sides by -2.

$$L=\frac{-\frac{15}{2}}{-2}$$

Express \frac{-\frac{15}{2}}{-2} as a single fraction.

$$L=\frac{-15}{2\left(-2\right)}$$

Multiply 2 and -2 to get -4.

$$L=\frac{-15}{-4}$$

Fraction \frac{-15}{-4} can be simplified to \frac{15}{4} by removing the negative sign from both the numerator and the denominator.

$$L=\frac{15}{4}$$

The system is now solved.

$$L=\frac{15}{4} \lambda =\frac{3}{8} K=\frac{23}{4}$$