B1. In implicit, explicit, and segment form, write the equation of the line $\mathrm{k}$ parallel to the line given by the equation $\frac{x}{4}+\frac{y}{-2}=1$, which passes through the intersection of the lines given by the equations $3 x+5 y=-3$ and $5 x+4 y=8$.
B1. Let's solve the system of equations $3x + 5y = -3$ and $5x + 4y = 8$ and we get $x = 4$ and $y = -3$. The intersection point of these two lines is $P(4, -3)$. The line $\frac{x}{4} + \frac{y}{-2} = 1$ can be written in explicit form as $y = \frac{1}{2}x - 2$, and its slope is $k = \frac{1}{2}$. Since parallel lines have the same slope, $k = \frac{1}{2}$ is also the slope of the desired line through the point $P(4, -3)$. The desired line in explicit form is given by the equation $y = \frac{1}{2}x - 5$, in implicit form as $-x + 2y + 10 = 0$, and in intercept form as $\frac{x}{10} + \frac{y}{-5} = 1$.

Note: The slope of the line can also be obtained from the intercept form in another way: from the intercepts, we get the intersection points on the coordinate axes $M(4, 0)$ and $N(0, -2)$, and using $k = \frac{y_2 - y_1}{x_2 - x_1}$, we again get $k = \frac{1}{2}$.

Solving the system of equations ....................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=57&width=1727&top_left_y=1462&top_left_x=176)

Converting to explicit form ....................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=57&width=1727&top_left_y=1568&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=51&width=1731&top_left_y=1619&top_left_x=171)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=57&width=1727&top_left_y=1668&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=52&width=1727&top_left_y=1716&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_9fbd0ae098c18b73b6bfg-21.jpg?height=60&width=1719&top_left_y=1763&top_left_x=174)