1. The problem is to complete the table for linear equations in two variables by converting from standard form to slope-intercept form, finding the slope, x-intercept, and y-intercept. 2. The standard form of a linear equation is $$Ax + By = C$$. 3. To convert to slope-intercept form $$y = mx + b$$, solve for $$y$$: $$By = -Ax + C$$ $$y = -\frac{A}{B}x + \frac{C}{B}$$ Here, slope $$m = -\frac{A}{B}$$ and y-intercept $$b = \frac{C}{B}$$. 4. The x-intercept is found by setting $$y=0$$: $$Ax + B(0) = C \Rightarrow x = \frac{C}{A}$$. 5. The y-intercept is found by setting $$x=0$$: $$A(0) + By = C \Rightarrow y = \frac{C}{B}$$. 6. Applying these rules to each equation: - 1. $$2x - y = -12$$ Slope-intercept: $$y = 2x + 12$$ Slope: $$m=2$$ X-intercept: $$2x = -12 \Rightarrow x = -6$$ (corrected to $(-6,0)$) Y-intercept: $$y = 12$$ (0,12) - 2. $$21x + 20y = 420$$ Slope-intercept: $$y = -\frac{21}{20}x + 21$$ Slope: $$m = -\frac{21}{20}$$ X-intercept: $$21x = 420 \Rightarrow x = 20$$ (20,0) Y-intercept: $$y = 21$$ (0,21) - 3. $$-7x + 18y = 126$$ Slope-intercept: $$y = \frac{7}{18}x + 7$$ Slope: $$m = \frac{7}{18}$$ X-intercept: $$-7x = 126 \Rightarrow x = -18$$ (-18,0) Y-intercept: $$y = 7$$ (0,7) - 4. $$7x + 8y = 56$$ Slope-intercept: $$y = -\frac{7}{8}x + 7$$ Slope: $$m = -\frac{7}{8}$$ X-intercept: $$7x = 56 \Rightarrow x = 8$$ (8,0) Y-intercept: $$y = 7$$ (0,7) - 5. $$-11x + 10y = 110$$ Slope-intercept: $$y = \frac{11}{10}x + 11$$ Slope: $$m = \frac{11}{10}$$ X-intercept: $$-11x = 110 \Rightarrow x = -10$$ (-10,0) Y-intercept: $$y = 11$$ (0,11) - 6. $$7x + 8y = -56$$ Slope-intercept: $$y = -\frac{7}{8}x - 7$$ Slope: $$m = -\frac{7}{8}$$ X-intercept: $$7x = -56 \Rightarrow x = -8$$ (-8,0) Y-intercept: $$y = -7$$ (0,-7) - 7. $$5x - 4y = 100$$ Slope-intercept: $$-4y = -5x + 100 \Rightarrow y = \frac{5}{4}x - 25$$ Slope: $$m = \frac{5}{4}$$ X-intercept: $$5x = 100 \Rightarrow x = 20$$ (20,0) Y-intercept: $$-4y = 100 \Rightarrow y = -25$$ (0,-25) - 8. $$-3x + y = 3$$ Slope-intercept: $$y = 3x + 3$$ Slope: $$m = 3$$ X-intercept: $$-3x = 3 \Rightarrow x = -1$$ (-1,0) Y-intercept: $$y = 3$$ (0,3) - 9. $$2x - 3y = 6$$ Slope-intercept: $$-3y = -2x + 6 \Rightarrow y = \frac{2}{3}x - 2$$ Slope: $$m = \frac{2}{3}$$ X-intercept: $$2x = 6 \Rightarrow x = 3$$ (3,0) Y-intercept: $$-3y = 6 \Rightarrow y = -2$$ (0,-2) - 10. $$7x + 5y = -70$$ Slope-intercept: $$5y = -7x - 70 \Rightarrow y = -\frac{7}{5}x - 14$$ Slope: $$m = -\frac{7}{5}$$ X-intercept: $$7x = -70 \Rightarrow x = -10$$ (-10,0) Y-intercept: $$y = -14$$ (0,-14) - 11. $$-7x + 6y = 126$$ Slope-intercept: $$6y = 7x + 126 \Rightarrow y = \frac{7}{6}x + 21$$ Slope: $$m = \frac{7}{6}$$ X-intercept: $$-7x = 126 \Rightarrow x = -18$$ (-18,0) Y-intercept: $$y = 21$$ (0,21) Final answers match the table given, except the x-intercept of problem 1 corrected to $(-6,0)$.