23) If a negative slope line passes through a point \( (4,8) \) and \( x \) intercept is \( 1 / 4 \) of the \( y \)-intercept. Find equation of straight line \begin{tabular}{l|l|}\hline a) \( 4 x+y=\mathbf{2 4} \) & b) \( 4 x-y=24 \) \\ \hline c) \( X+4 y=24 \) & d) \( X-4 y=24 \) \\ \hline\end{tabular}

To find the equation of the straight line with a negative slope that passes through the point \( (4,8) \) and where the \( x \)-intercept is \( \frac{1}{4} \) of the \( y \)-intercept, we can start by denoting the \( y \)-intercept as \( b \). The \( x \)-intercept will then be \( \frac{b}{4} \). The slope \( m \) can be determined from the intercepts: \( m = -\frac{b}{\frac{b}{4}} = -4 \). Therefore, the equation in intercept form is: \[ y = -4x + b. \] Since we know it passes through \( (4,8) \), we can plug that into the equation to solve for \( b \): \[ 8 = -4(4) + b \implies 8 = -16 + b \implies b = 24. \] Thus, the equation becomes: \[ y = -4x + 24. \] Rearranging this gives us: \[ 4x + y = 24. \] So the correct answer is \( \text{(a) } 4x + y = 24 \). In the world of mathematics, understanding the connections between intercepts and slopes can help you quickly find equations of lines! This technique allows you to graph a line just by knowing two key points—perfect for quickly sketching out graphs on tests or homework. If you're ever unsure about slope-intercept forms or how to rearrange equations, a common pitfall is to confuse the signs. Remember, a negative slope means the line goes down as you move to the right! Practice identifying slopes on graphs to solidify your understanding.