Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=-4 x-2,(5,3)$$

Short answer

The equation of the line parallel to the given line and passing through the point (5, 3) is y = -4x + 23.

Step-by-step solution

Identifying the slope

First, identify the slope of the given line. Looking at the equation y = -4x - 2, we can identify the coefficient of x, -4, as the slope of the line. Because parallel lines share the same slope, we know that the slope of the line that we need to find is also -4.

Using the point-slope formula

Next, use the point-slope formula, y - y1 = m(x - x1), to write an equation for the new line. The slope \(m\) is -4, given by the equation. The point (x1, y1) we use is (5, 3), which was given as a point that lies on our new line.

Substituting values into the formula

Now, substitute the values into the point-slope formula. This results in the equation y - 3 = -4(x - 5).

Simplifying the equation

Lastly, simplify the equation to its standard slope-intercept form y = mx + c. By simplifying y - 3 = -4(x - 5), distribute -4 over (x - 5) to get -4x + 20. Bring the 3 over to the other side to get the final equation, y = -4x + 23.