Question

1. identify which lines are parallel: $y = -\frac{1}{2}x + 3$; $y = \frac{1}{2}x + 1$; $y = 2x$; $x + 2y = 4$.

Explanation:

Step1: Recall Parallel Line Condition

Parallel lines have equal slopes. The slope-intercept form is \(y = mx + b\), where \(m\) is the slope. For a line in standard form \(Ax + By = C\), slope \(m = -\frac{A}{B}\).

Step2: Find Slope of Each Line

• Line 1: \(y = -\frac{1}{2}x + 3\), slope \(m_1 = -\frac{1}{2}\).
• Line 2: \(y = \frac{1}{2}x + 1\), slope \(m_2 = \frac{1}{2}\).
• Line 3: \(y = 2x\), slope \(m_3 = 2\).
• Line 4: \(x + 2y = 4\) → \(2y = -x + 4\) → \(y = -\frac{1}{2}x + 2\), slope \(m_4 = -\frac{1}{2}\).

Step3: Compare Slopes

\(m_1 = -\frac{1}{2}\) and \(m_4 = -\frac{1}{2}\), so their slopes are equal.

Answer:

The lines \(y = -\frac{1}{2}x + 3\) and \(x + 2y = 4\) are parallel.