Question

solve the equation.
\sqrt{5 - 7t} = 9
suggested tutorial:
learn it: solve radical equations.
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Explanation:

Step1: Square both sides to eliminate radical

To solve the equation \(\sqrt{5 - 7T}=9\), we first square both sides of the equation. Squaring the left side will eliminate the square root, and squaring the right side will give us a new equation to work with.

$$ (\sqrt{5 - 7T})^2 = 9^2 $$

Simplifying both sides, we get:

$$ 5 - 7T = 81 $$

Step2: Solve for T

Now we solve the linear equation \(5 - 7T = 81\) for \(T\). First, subtract 5 from both sides:

$$ -7T = 81 - 5 $$

$$ -7T = 76 $$

Then, divide both sides by -7:

$$ T = \frac{76}{-7}=-\frac{76}{7} $$

Wait, let's check the calculation again. Wait, 81 - 5 is 76? Wait, no, 81 - 5 is 76? Wait, 81 - 5 is 76? Wait, 81 minus 5 is 76? Wait, 80 - 5 is 75, plus 1 is 76. Yes. Then dividing by -7: \(T = -\frac{76}{7}\approx - 10.857\). But let's check the original equation with \(T = 4\) (the suggested answer which was wrong). Plugging \(T = 4\) into the left side: \(\sqrt{5-7\times4}=\sqrt{5 - 28}=\sqrt{-23}\), which is not a real number, so that's why it was wrong. Let's redo the steps carefully.

Step1: Square both sides

Given \(\sqrt{5 - 7T}=9\). Square both sides:

$$ (\sqrt{5 - 7T})^2=9^2 $$

Which simplifies to:

$$ 5 - 7T = 81 $$

Step2: Isolate the term with T

Subtract 5 from both sides:

$$ -7T=81 - 5 $$

$$ -7T = 76 $$

Wait, no, 81 - 5 is 76? Wait, 81 - 5: 81 - 5 = 76. Then divide both sides by -7:

$$ T=\frac{76}{-7}=-\frac{76}{7}\approx - 10.86 $$

But let's check if this is correct. Plug \(T = -\frac{76}{7}\) back into the original equation:
Left side: \(\sqrt{5-7\times(-\frac{76}{7})}=\sqrt{5 + 76}=\sqrt{81}=9\), which matches the right side. So the correct solution is \(T = -\frac{76}{7}\).

Wait, maybe I made a mistake in the first check with \(T = 4\). Let's check \(T = 4\): \(\sqrt{5-7\times4}=\sqrt{5 - 28}=\sqrt{-23}\), which is imaginary, so that's invalid. So the correct solution is \(T = -\frac{76}{7}\).

Answer:

\(T = -\dfrac{76}{7}\) (or approximately \(T\approx - 10.86\))