Solve the following equation in x: $$ 2(3x - 5) + 8 = 4x - 5 $$ Write $$\frac{3^{5}}{3^{6}}$$ in the form $$3^{k}$$, where $$k \in \mathbb{R}$$ - Leaving Cert Mathematics - Question 2 - 2023

To simplify the fraction, use the property of exponents:

$\frac{a^{m}}{a^{n}} = a^{m-n}$

Applying this to our problem:

$\frac{3^{5}}{3^{6}} = 3^{5 - 6} = 3^{-1}$

Thus, $k = -1$.

We have the following equations:

1. $3x + 2y = 1$
2. $7x + 5y = -2$

First, let's solve Equation 1 for y:

$$2y = 1 - 3x$$

Thus:

y = \frac{1 - 3x}{2}$$ Now substitute this into Equation 2:

7x + 5\left(\frac{1 - 3x}{2}\right) = -2

$$Multiply through by 2 to eliminate the fraction:$$

14x + 5(1 - 3x) = -4

$$Distributing gives:$$

14x + 5 - 15x = -4

$$Combine like terms:$$

-x + 5 = -4

$$Moving 5 to the right side:$$

-x = -4 - 5 \ -x = -9\

$$So:$$

x = 9$$

Now substitute the value of x back into our equation for y:

y = \frac{1 - 3(9)}{2}$$ Thus:

y = \frac{1 - 27}{2} = -13$$

The solutions are:

$x = 9$ and $y = -13$.