What is the equation of the line that is parallel to the line [tex]5x + 2y = 12[/tex] and passes through the point [tex](-2, 4)[/tex]?

A. [tex]y = -\frac{5}{2}x - 1[/tex]

B. [tex]y = -\frac{5}{2}x + 5[/tex]

C. [tex]y = \frac{2}{5}x - 1[/tex]

D. [tex]y = \frac{2}{5}x + 5[/tex]



Answer :

To find the equation of the line that is parallel to the given line and passes through a specific point, we can follow these steps:

1. Identify the slope of the given line: The given line is \(5x + 2y = 12\). We need to find its slope. To do this, we convert it to slope-intercept form \(y = mx + b\) where \(m\) is the slope.

Let's rewrite the line equation \(5x + 2y = 12\) in slope-intercept form:
[tex]\[ 2y = -5x + 12 \][/tex]
[tex]\[ y = -\frac{5}{2}x + 6 \][/tex]
From this, we see that the slope \(m\) of the given line is \(-\frac{5}{2}\).

2. Determine the slope of the parallel line: Since parallel lines have the same slope, the slope of our new line will also be \(-\frac{5}{2}\).

3. Use the point-slope form of the line equation: We know the slope and a point \((-2, 4)\) through which the line passes. The point-slope form of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting \(m = -\frac{5}{2}\), \(x_1 = -2\), and \(y_1 = 4\) into the point-slope form:
[tex]\[ y - 4 = -\frac{5}{2}(x - (-2)) \][/tex]
Simplify:
[tex]\[ y - 4 = -\frac{5}{2}(x + 2) \][/tex]
[tex]\[ y - 4 = -\frac{5}{2}x - 5 \][/tex]
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]

So, the equation of the line that is parallel to \(5x + 2y = 12\) and passes through the point \((-2, 4)\) is:
[tex]\[ y = -\frac{5}{2}x - 1 \][/tex]

Hence, the correct answer is:
[tex]\[y=-\frac{5}{2} x-1\][/tex]