For the linear equation \( 3 x+2 y-12=0 \), the \( x \)-intercept is \( \square \) and the \( y \)-intercept is \( \square \). The slope of the graph of this equation is \( \square \). The equation in slope-intercept form is Need Help? Read it

To find the \( x \)-intercept, set \( y = 0 \) in the equation \( 3x + 2y - 12 = 0\). This gives \( x = 4 \), so the \( x \)-intercept is \( 4 \). For the \( y \)-intercept, set \( x = 0 \) to get \( y = 6 \), leading to a \( y \)-intercept of \( 6 \). The slope \( m \) can be derived from the equation in slope-intercept form \( y = mx + b \). Rearranging gives \( y = -\frac{3}{2}x + 6 \), so the slope is \( -\frac{3}{2} \). To convert to slope-intercept form, isolate \( y \). Begin with your equation \( 3x + 2y - 12 = 0 \). Rearranging leads to \( 2y = -3x + 12 \), then dividing by 2 gives us \( y = -\frac{3}{2}x + 6 \). Voilà! The slope-intercept form is ready for action! The \( x \)-intercepts and \( y \)-intercepts are key points that graph the linear equation's path. They provide a quick, visual way to understand how the line behaves on a coordinate plane. Plotting these points can kickstart your journey through the Cartesian landscape of graphs and equations! For practicing your skills, try changing the coefficients in your original equation. Each adjustment opens a new set of intercepts and slopes to explore, allowing you to develop a deeper understanding of linear relationships. Challenge a friend to see who can derive the intercepts and slope the fastest!