If #4l - 3m = 15# and lm = 10, then find the value of #16l^2 + 9m^2#?

Question

Savannah Anderson · Accepted Answer

#l = 10/m#

#4(10/m) - 3m = 15#

#40/m - 3m = 15#

#40 - 3m^2 = 15m#

#0 = 3m^2 + 15m - 40#

By the quadratic formula:

#m = (-15 +- sqrt(15^2 - (4 xx 3 xx -40)))/(2 xx 3)#

#m = (-15 +- sqrt(705))/6#

#l = 10/((-15 +- sqrt(705))/6)#

#l = 60/(-15 +- sqrt(705))#

Now, we can do our calculation:

#16(60/(-15 +- sqrt(705)))^2 + 9((-15 +- sqrt(705))/6)^2#

I will leave the rest of the simplification up to you.

Hopefully this helps!

Avery Alexander · Answer

undefined To find the value of $ 16l^2 + 9m^2 $, first, we need to express $ l $ and $ m $ in terms of each other using the given equations.

From $ lm = 10 $, we can express $ m $ in terms of $ l $ as $ m = \frac{10}{l} $.

Now, substitute this expression for $ m $ into the equation $ 4l - 3m = 15 $:
$$ 4l - 3\left(\frac{10}{l}\right) = 15 $$

Solve for $ l $:
$$ 4l - \frac{30}{l} = 15 $$
$$ 4l^2 - 30 = 15l $$
$$ 4l^2 - 15l - 30 = 0 $$

Now, solve this quadratic equation to find the values of $ l $. Once you have the values of $ l $, you can find the corresponding values of $ m $ using $ m = \frac{10}{l} $.

After finding the values of $ l $ and $ m $, substitute them into the expression $ 16l^2 + 9m^2 $ to find the final answer.