If you consider the logarithm base 3 of both sides, you have
You can use a rule of logarithms that allow you to turn exponents into multiplicative factors:
So the equation becomes
Now, by definition, you have
So, you have
So, the equation becomes
Let r be the lesser root and r^2 be the greater.......the sum of the roots = -b/a = -[-6] / 1 = 6
So we have that
r^2 + r = 6 → r^2 + r - 6 = 0
Factor
(r + 3) (r - 2) = 0
So r = -3 or r = 2
Then r^2 = 9 or r^2 =4
And the product of the roots = c/a = k/a = k
So....k = (-3)(9) = -27 or k = (2)(4) = 8
Check
x^2 - 6x - 27 = 0 factors as (x + 3)(x - 9) = 0 and the roots are -3 and 9
x^2 - 6x + 8 = 0 factors as (x -2) (x - 4) =0 and the roots are 2 and 4
Answer: (x, y) = (2, 3)
This is the system of equations 2x - 3y = -5, 5x - 2y = 4. Multiply the first equation by 2 and the second equation by 3 to get 4x - 6y = -10, 15x - 6y = 12. Now we can use elimination: subtract the equations to get -11x = -22, so x = 2. 2x - 3y = 2(2) - 3y = 4 - 3y = -5, so 3y = 9 and y = 3. The solution is (2, 3).
i hope this helped! :D
Answer:
<u>x = 18</u>
Step-by-step explanation:
1. Let's solve for x, using the supplementary angles theorem, this way:
∠ C = 180 - 7x + 2 because they are supplementary angles. And these angles add up to 180 degrees.
∠ D = 180 - 9x + 31 because they are supplementary angles. And these angles add up to 180 degrees.
∠ E = 180 - 4x - 33 because they are supplementary angles. And these angles add up to 180 degrees.
Now, let's recall that the interior angles of a triangle add up to 180 degrees. Thus:
∠ C + ∠ D + ∠ E = 180 °
Replacing with the real values for solving for x:
180 - 9x + 31 + 180 - 4x - 33 + 180 - 7x + 2 = 180
- 20 x + 540 = 180
- 20 x = 180 - 540 (Subtracting 540 to both sides)
- 20 x = - 360
<u>x = 18 (Dividing by - 20)</u>
<u>Proving that x = 18 is correct</u>
180 - 9x + 31 + 180 - 4x - 33 + 180 - 7x + 2 = 180
180 - 9 (18) + 31 + 180 - 4 (18) - 33 + 180 - 7 (18) + 2 = 180
180 - 162 + 31 + 180 - 72 - 33 + 180 - 126 + 2 = 180
573 - 393 = 180
180 = 180
<u>We proved that x = 18 is correct</u>
