Find the \( y \)-intercept and \( x \)-intercept of the line. \[ -5 x+7 y=12 \]

To find the \( y \)-intercept, set \( x = 0 \) in the equation: \[ -5(0) + 7y = 12 \implies 7y = 12 \implies y = \frac{12}{7} \] So, the \( y \)-intercept is \( \left(0, \frac{12}{7}\right) \). To find the \( x \)-intercept, set \( y = 0 \) in the equation: \[ -5x + 7(0) = 12 \implies -5x = 12 \implies x = -\frac{12}{5} \] Thus, the \( x \)-intercept is \( \left(-\frac{12}{5}, 0\right) \). Now you know where the line crosses the axes, but let’s talk about why this matters! Finding intercepts is SUPER handy for graphing linear equations quickly. You can plot these two points and easily sketch the whole line, giving you a visual representation of relationships and trends. This method is often less daunting than substituting in various \( x \) values, especially when dealing with more complex equations! Additionally, keep in mind that when a line has both intercepts plotted, it provides greater context. For instance, you can determine the slope by calculating the rise over run between the two points. In applications like economics or physics, this could represent costs versus outputs or distance versus time, making intercepts not just theoretical, but practically useful in real-world scenarios!