Solve (1/2(10)(1)^2)(7)=(1/2(10)(1)^2+(0.2)(0.5)^2)omega_{f}

$$( \frac { 1 } { 2 } ( 10 ) ( 1 ) ^ { 2 } ) ( 7 ) = ( \frac { 1 } { 2 } ( 10 ) ( 1 ) ^ { 2 } + ( 0.2 ) ( 0.5 ) ^ { 2 } ) \omega _ { f }$$

Solve for ω_f

$$\omega _{f}=\frac{700}{101}\approx 6.930693069$$

Steps for Solving Linear Equation

Multiply \frac{1}{2} and 10 to get \frac{10}{2}.

$$\frac{10}{2}\times 1^{2}\times 7=\left(\frac{1}{2}\times 10\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Divide 10 by 2 to get 5.

$$5\times 1^{2}\times 7=\left(\frac{1}{2}\times 10\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Calculate 1 to the power of 2 and get 1.

$$5\times 1\times 7=\left(\frac{1}{2}\times 10\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Multiply 5 and 1 to get 5.

$$5\times 7=\left(\frac{1}{2}\times 10\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Multiply 5 and 7 to get 35.

$$35=\left(\frac{1}{2}\times 10\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Multiply \frac{1}{2} and 10 to get \frac{10}{2}.

$$35=\left(\frac{10}{2}\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Divide 10 by 2 to get 5.

$$35=\left(5\times 1^{2}+0.2\times 0.5^{2}\right)\omega _{f}$$

Calculate 1 to the power of 2 and get 1.

$$35=\left(5\times 1+0.2\times 0.5^{2}\right)\omega _{f}$$

Multiply 5 and 1 to get 5.

$$35=\left(5+0.2\times 0.5^{2}\right)\omega _{f}$$

Calculate 0.5 to the power of 2 and get 0.25.

$$35=\left(5+0.2\times 0.25\right)\omega _{f}$$

Multiply 0.2 and 0.25 to get 0.05.

$$35=\left(5+0.05\right)\omega _{f}$$

Add 5 and 0.05 to get 5.05.

$$35=5.05\omega _{f}$$

Swap sides so that all variable terms are on the left hand side.

$$5.05\omega _{f}=35$$

Divide both sides by 5.05.

$$\omega _{f}=\frac{35}{5.05}$$

Expand \frac{35}{5.05} by multiplying both numerator and the denominator by 100.

$$\omega _{f}=\frac{3500}{505}$$

Reduce the fraction \frac{3500}{505} to lowest terms by extracting and canceling out 5.

$$\omega _{f}=\frac{700}{101}$$