1. The problem asks to complete the chart by finding the x-intercept, y-intercept, and slope-intercept form $y=mx+b$ for each equation. 2. Recall the intercepts: - The x-intercept is where $y=0$. - The y-intercept is where $x=0$. 3. For slope-intercept form, solve the equation for $y$ to get $y=mx+b$ where $m$ is the slope and $b$ is the y-intercept. --- **a) Equation:** $2x + 3y = 12$ - Find x-intercept: set $y=0$ $$2x + 3(0) = 12 \Rightarrow 2x = 12 \Rightarrow x = \frac{12}{2} = 6$$ - Find y-intercept: set $x=0$ $$2(0) + 3y = 12 \Rightarrow 3y = 12 \Rightarrow y = \frac{12}{3} = 4$$ - Solve for $y$ to get slope-intercept form: $$2x + 3y = 12 \Rightarrow 3y = 12 - 2x \Rightarrow y = \frac{12 - 2x}{3} = -\frac{2}{3}x + 4$$ --- **b) Equation:** $4x - 6y = 24$ - Find x-intercept: set $y=0$ $$4x - 6(0) = 24 \Rightarrow 4x = 24 \Rightarrow x = \frac{24}{4} = 6$$ - Find y-intercept: set $x=0$ $$4(0) - 6y = 24 \Rightarrow -6y = 24 \Rightarrow y = \frac{24}{-6} = -4$$ - Solve for $y$: $$4x - 6y = 24 \Rightarrow -6y = 24 - 4x \Rightarrow y = \frac{4x - 24}{6} = \frac{4}{6}x - 4 = \frac{2}{3}x - 4$$ --- **c) Equation given in slope-intercept form:** $y = -4x + 5$ - Find x-intercept: set $y=0$ $$0 = -4x + 5 \Rightarrow 4x = 5 \Rightarrow x = \frac{5}{4}$$ - Find y-intercept: set $x=0$ $$y = -4(0) + 5 = 5$$ - Convert to standard form: $$y = -4x + 5 \Rightarrow 4x + y = 5$$ --- **d) Equation given in slope-intercept form:** $y = \frac{2}{3}x - 4$ - Find x-intercept: set $y=0$ $$0 = \frac{2}{3}x - 4 \Rightarrow \frac{2}{3}x = 4 \Rightarrow x = 4 \times \frac{3}{2} = 6$$ - Find y-intercept: set $x=0$ $$y = \frac{2}{3}(0) - 4 = -4$$ - Convert to standard form: $$y = \frac{2}{3}x - 4 \Rightarrow 3y = 2x - 12 \Rightarrow 2x - 3y = 12$$ --- **Final chart:** | Standard Form | X-Intercept | Y-Intercept | y=mx+b form | |-----------------|-------------|-------------|-------------------| | a) 2x + 3y = 12 | 6 | 4 | $y = -\frac{2}{3}x + 4$ | | b) 4x - 6y = 24 | 6 | -4 | $y = \frac{2}{3}x - 4$ | | c) 4x + y = 5 | $\frac{5}{4}$ | 5 | $y = -4x + 5$ | | d) 2x - 3y = 12 | 6 | -4 | $y = \frac{2}{3}x - 4$ |