1. **Determine if the lines are parallel (\(\parallel\)), perpendicular (\(\perp\)), or neither (N).** Recall: - Lines are parallel if their slopes are equal. - Lines are perpendicular if the product of their slopes is \(-1\). - Otherwise, they are neither. **a.** \(y=4x\), \(y=4x-4\) - Both slopes are 4. - Since slopes are equal, lines are parallel: \(\parallel\). **b.** \(y=\frac{1}{2}x-2\), \(y=2x\) - Slopes: \(\frac{1}{2}\) and 2. - Product: \(\frac{1}{2} \times 2 = 1 \neq -1\). - Not perpendicular, slopes not equal, so neither: N. **c.** \(y=-x\), \(y=x-1\) - Slopes: \(-1\) and 1. - Product: \(-1 \times 1 = -1\), so perpendicular: \(\perp\). **d.** \(y=x+7\), \(y=x-7\) - Both slopes 1. - Parallel: \(\parallel\). **e.** \(y=\frac{3}{4}x+5\), \(y=\frac{4}{3}x-8\) - Slopes: \(\frac{3}{4}\) and \(\frac{4}{3}\). - Product: \(\frac{3}{4} \times \frac{4}{3} = 1 \neq -1\). - Neither: N. **f.** \(y=3x+2\), \(y=-\frac{1}{3}x\) - Slopes: 3 and \(-\frac{1}{3}\). - Product: \(3 \times -\frac{1}{3} = -1\), perpendicular: \(\perp\). **g.** \(y=\frac{1}{4}x-2\), \(y=\frac{1}{4}x+3\) - Both slopes \(\frac{1}{4}\). - Parallel: \(\parallel\). **h.** \(y=2x+5\), \(4x - 2y + 6 = 0\) - Rearrange second: \(4x - 2y + 6 = 0 \Rightarrow -2y = -4x - 6 \Rightarrow y = 2x + 3\) - Both slopes 2. - Parallel: \(\parallel\). **i.** \(x + y = 4\), \(y = x - 3\) - First: \(y = -x + 4\), slope \(-1\). - Second slope 1. - Product \(-1 \times 1 = -1\), perpendicular: \(\perp\). **j.** \(y=\frac{1}{2}x - 4\), \(x - 2y + 1 = 0\) - Rearrange second: \(x - 2y + 1 = 0 \Rightarrow -2y = -x - 1 \Rightarrow y = \frac{1}{2}x + \frac{1}{2}\) - Both slopes \(\frac{1}{2}\). - Parallel: \(\parallel\). **k.** \(y=6\), \(y=2\) - Both horizontal lines, slope 0. - Parallel: \(\parallel\). **l.** \(y=4\), \(y=-4\) - Both horizontal, slope 0. - Parallel: \(\parallel\). **m.** \(y=7\), \(x=-7\) - First horizontal (slope 0), second vertical (undefined slope). - Horizontal and vertical lines are perpendicular: \(\perp\). **n.** \(x=\frac{1}{2}\), \(x=-\frac{1}{2}\) - Both vertical lines. - Parallel: \(\parallel\). **o.** \(y=5\), \(x=7\) - Horizontal and vertical lines. - Perpendicular: \(\perp\). **n.** (second n) \(y=x\), \(y=2x\) - Slopes 1 and 2. - Product \(1 \times 2 = 2 \neq -1\), not equal. - Neither: N. **p.** \(y=\frac{1}{3}x\), \(y=-3x\) - Slopes \(\frac{1}{3}\) and \(-3\). - Product \(\frac{1}{3} \times -3 = -1\), perpendicular: \(\perp\). **q.** \(y=0.5x\), \(y=\frac{1}{2}x\) - Both slopes \(\frac{1}{2}\). - Parallel: \(\parallel\). **r.** \(y=-\frac{2}{3}x\), \(y=\frac{3}{2}x\) - Product \(-\frac{2}{3} \times \frac{3}{2} = -1\), perpendicular: \(\perp\). **s.** \(y=4\), \(y=2x\) - Slopes 0 and 2. - Product 0, not -1, slopes not equal. - Neither: N. --- 2. **State the equations of the labelled lines in the graph:** a. Horizontal line at \(y=5\) b. Horizontal line at \(y=-7\) c. Horizontal line at \(y=-6\) d. Vertical line at \(x=2\) e. Diagonal line through origin with negative slope crossing y-axis at 0, descending top-left to bottom-right, equation \(y=2x\) given (note: negative slope stated but equation positive slope, assuming given equation is correct)\ --- **Final answers:** a. \(\parallel\) b. N c. \(\perp\) d. \(\parallel\) e. N f. \(\perp\) g. \(\parallel\) h. \(\parallel\) i. \(\perp\) j. \(\parallel\) k. \(\parallel\) l. \(\parallel\) m. \(\perp\) n. \(\parallel\) o. \(\perp\) n. N p. \(\perp\) q. \(\parallel\) r. \(\perp\) s. N