Solve frac{20^{2}+(1.5t)^{2}-(12+1.5t)^2}{60t}=34^2+(1.5t)^2-(30+1.5t)^2/102t

$$\frac { 20 ^ { 2 } + ( 1.5 t ) ^ { 2 } - ( 12 + 1.5 t ) ^ { 2 } } { 60 t } = \frac { 34 ^ { 2 } + ( 1.5 t ) ^ { 2 } - ( 30 + 1.5 t ) ^ { 2 } } { 102 t }$$

Solve for t

$$t = -\frac{56}{9} = -6\frac{2}{9} \approx -6.222222222$$

Steps for Solving Linear Equation

Variable t cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1020t, the least common multiple of 60t,102t.

$$17\left(20^{2}+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 20 to the power of 2 and get 400.

$$17\left(400+\left(1.5t\right)^{2}-\left(12+1.5t\right)^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$17\left(400+1.5^{2}t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$17\left(400+2.25t^{2}-\left(12+1.5t\right)^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12+1.5t\right)^{2}.

$$17\left(400+2.25t^{2}-\left(144+36t+2.25t^{2}\right)\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

To find the opposite of 144+36t+2.25t^{2}, find the opposite of each term.

$$17\left(400+2.25t^{2}-144-36t-2.25t^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Subtract 144 from 400 to get 256.

$$17\left(256+2.25t^{2}-36t-2.25t^{2}\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$17\left(256-36t\right)=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Use the distributive property to multiply 17 by 256-36t.

$$4352-612t=10\left(34^{2}+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 34 to the power of 2 and get 1156.

$$4352-612t=10\left(1156+\left(1.5t\right)^{2}-\left(30+1.5t\right)^{2}\right)$$

Expand \left(1.5t\right)^{2}.

$$4352-612t=10\left(1156+1.5^{2}t^{2}-\left(30+1.5t\right)^{2}\right)$$

Calculate 1.5 to the power of 2 and get 2.25.

$$4352-612t=10\left(1156+2.25t^{2}-\left(30+1.5t\right)^{2}\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(30+1.5t\right)^{2}.

$$4352-612t=10\left(1156+2.25t^{2}-\left(900+90t+2.25t^{2}\right)\right)$$

To find the opposite of 900+90t+2.25t^{2}, find the opposite of each term.

$$4352-612t=10\left(1156+2.25t^{2}-900-90t-2.25t^{2}\right)$$

Subtract 900 from 1156 to get 256.

$$4352-612t=10\left(256+2.25t^{2}-90t-2.25t^{2}\right)$$

Combine 2.25t^{2} and -2.25t^{2} to get 0.

$$4352-612t=10\left(256-90t\right)$$

Use the distributive property to multiply 10 by 256-90t.

$$4352-612t=2560-900t$$

Add 900t to both sides.

$$4352-612t+900t=2560$$

Combine -612t and 900t to get 288t.

$$4352+288t=2560$$

Subtract 4352 from both sides.

$$288t=2560-4352$$

Subtract 4352 from 2560 to get -1792.

$$288t=-1792$$

Divide both sides by 288.

$$t=\frac{-1792}{288}$$

Reduce the fraction \frac{-1792}{288} to lowest terms by extracting and canceling out 32.

$$t=-\frac{56}{9}$$