Solve 5^2-a^2=7^2-(8-a)^2

$$5 ^ { 2 } - a ^ { 2 } = 7 ^ { 2 } - ( 8 - a ) ^ { 2 }$$

Solve for a

$$a = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps for Solving Linear Equation

Calculate 5 to the power of 2 and get 25.

$$25-a^{2}=7^{2}-\left(8-a\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$25-a^{2}=49-\left(8-a\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-a\right)^{2}.

$$25-a^{2}=49-\left(64-16a+a^{2}\right)$$

To find the opposite of 64-16a+a^{2}, find the opposite of each term.

$$25-a^{2}=49-64+16a-a^{2}$$

Subtract 64 from 49 to get -15.

$$25-a^{2}=-15+16a-a^{2}$$

Subtract 16a from both sides.

$$25-a^{2}-16a=-15-a^{2}$$

Add a^{2} to both sides.

$$25-a^{2}-16a+a^{2}=-15$$

Combine -a^{2} and a^{2} to get 0.

$$25-16a=-15$$

Subtract 25 from both sides.

$$-16a=-15-25$$

Subtract 25 from -15 to get -40.

$$-16a=-40$$

Divide both sides by -16.

$$a=\frac{-40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out -8.

$$a=\frac{5}{2}$$