To solve the problem, we need to find the equation of the line parallel to $y = \frac{1}{2}x - 1$ that passes through the point $(6, -5)$.

Step 1: Identify the slope of the given line

The slope-intercept form of a line is: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

From the given equation $y = \frac{1}{2}x - 1$, the slope $m$ is: $m = \frac{1}{2}$

Since parallel lines have the same slope, the slope of the required line is also: $m = \frac{1}{2}.$

Step 2: Use the point-slope form of the equation

The point-slope form of a line is: $y - y_1 = m(x - x_1),$ where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

Here:

• $m = \frac{1}{2}$
• $(x_1, y_1) = (6, -5)$

Substitute these values into the point-slope form: $y - (-5) = \frac{1}{2}(x - 6).$

Step 3: Simplify to slope-intercept form

Simplify the equation step by step:

1. Replace $y - (-5)$ with $y + 5$: $y + 5 = \frac{1}{2}(x - 6).$

2. Distribute $\frac{1}{2}$ to $x - 6$: $y + 5 = \frac{1}{2}x - 3.$

3. Subtract 5 from both sides to isolate $y$: $y = \frac{1}{2}x - 3 - 5.$

4. Combine like terms: $y = \frac{1}{2}x - 8.$

Final Answer:

The equation of the line in slope-intercept form is: $y = \frac{1}{2}x - 8.$

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Here are 5 related questions for practice:

1. Find the equation of a line parallel to $y = -3x + 7$ through the point $(4, 2)$.
2. Find the equation of a line perpendicular to $y = \frac{1}{2}x + 3$ passing through $(2, -1)$.
3. What is the slope of a line parallel to $y = -5x + 9$?
4. Write the equation of a line with slope $2$ passing through $(0, -4)$.
5. Find the y-intercept of a line parallel to $y = \frac{3}{4}x - 2$ through $(-2, 5)$.

Tip: For parallel lines, the slopes are always the same, but the y-intercepts may differ.