This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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To determine the relation between lines, we compare their slopes. The general form of a linear equation is y = mx + c, where m is the slope and c is the y-intercept. If two lines have the same slope (m_1 = m_2), they are parallel*. If the product of their slopes is -1 (m_1 · m_2 = -1), they are perpendicular*. 1) For the lines y = 3x + 1 and y = 3x + 5: Step 1: Identify the slopes of the two lines. The first line is y = 3x + 1, so its slope m_1 = 3. The second line is y = 3x + 5, so its slope m_2 = 3. Step 2: Compare the slopes. Since m_1 = 3 and m_2 = 3, the slopes are equal (m_1 = m_2). Step 3: State the relationship. The lines are parallel. Parallel 2) For the lines y = -(3)/(2)x + 1 and y = (2)/(3)x + 4: Step 1: Identify the slopes of the two lines. The first line is y = -(3)/(2)x + 1, so its slope m_1 = -(3)/(2). The second line is y = (2)/(3)x + 4, so its slope m_2 = (2)/(3). Step 2: Calculate the product of the slopes. m_1 · m_2 = (-(3)/(2)) · ((2)/(3)) m_1 · m_2 = -(3 × 2)/(2 × 3) m_1 · m_2 = -(6)/(6) m_1 · m_2 = -1 Step 3: State the relationship. Since the product of their slopes is -1, the lines are perpendicular. Perpendicular Send me the next one 📸