Questions: Write an equation for a line parallel to y=2x+5 and passing through the point (4,3) y=

Transcript text: Write an equation for a line parallel to $y=2 x+5$ and passing through the point $(4,3)$ \[ y=\square \]

Solution

Solution Steps

Step 1: Identify the Slope of the Given Line

The equation of the given line is $ y = 2x + 5 $. This is in the slope-intercept form $ y = mx + b $, where $ m $ is the slope. Therefore, the slope of the given line is $ m = 2 $.

Step 2: Determine the Slope of the Parallel Line

Lines that are parallel have the same slope. Therefore, the slope of the line parallel to the given line is also $ m = 2 $.

Step 3: Use the Point-Slope Form to Write the Equation

The point-slope form of a line is given by:

\[ y - y_1 = m(x - x_1) \]

where $ (x_1, y_1) $ is a point on the line. We are given the point $ (4, 3) $. Substituting the slope $ m = 2 $ and the point $ (4, 3) $ into the point-slope form, we have:

\[ y - 3 = 2(x - 4) \]

Step 4: Simplify the Equation

Distribute the slope on the right side:

\[ y - 3 = 2x - 8 \]

Add 3 to both sides to solve for $ y $:

\[ y = 2x - 8 + 3 \]

\[ y = 2x - 5 \]