Respuesta :
First find the slope of the given line like this:
[tex]5x + 2y = 12\\2y=-5x+12\\y=\frac{-5}{2}x+6[/tex]
If two lines are parallel, then they have the same slope, so the equation we are looking for is in the form:
[tex]y=\frac{-5}{2}x+a[/tex] and we have to find b using the given point like this:
Substitute x = -2 and y=4 in the above equation like this and solve for a:
[tex]4=\frac{-5}{2}(-2)+a[/tex]
[tex]4=5+a[/tex]
[tex]a=4-5=-1[/tex]
The equation of the line is then the following:
[tex]y=\frac{-5}{2}x-1.[/tex]
[tex]5x + 2y = 12\\2y=-5x+12\\y=\frac{-5}{2}x+6[/tex]
If two lines are parallel, then they have the same slope, so the equation we are looking for is in the form:
[tex]y=\frac{-5}{2}x+a[/tex] and we have to find b using the given point like this:
Substitute x = -2 and y=4 in the above equation like this and solve for a:
[tex]4=\frac{-5}{2}(-2)+a[/tex]
[tex]4=5+a[/tex]
[tex]a=4-5=-1[/tex]
The equation of the line is then the following:
[tex]y=\frac{-5}{2}x-1.[/tex]
The equation of the line which is parallel to the line [tex]5x+2y=12[/tex] and passes through the point [tex](-2,4)[/tex] is [tex]\fbox{\begin\\\ \math y=\frac{-5x}{2}-1\\\end{minispace}}[/tex].
Further explanation:
In the question it is given that two lines are parallel to each other. The equation of the first line is [tex]5x+2y=12[/tex] and the equation of the second line is needed to be determined.
As per the question the second line passes through the point [tex](-2,4)[/tex].
The equation of the first line is as follows:
[tex]5x+2y=12[/tex] (1)
Simplify the above equation into its slope intercept form.
Subtract [tex]5x[/tex] from equation (1).
[tex]\begin{aligned}5x+2y-5x&=-5x+12\\2y&=-5x+12\end{aligned}[/tex]
Divide the above equation by [tex]2[/tex].
[tex]\begin{aligned}\frac{2y}{2}&=\frac{-5x}{2}+\frac{12}{2}\\y&=\frac{-5x}{2}+6\end{aligned}[/tex] (2)
The general equation of the slope intercept form of the line is as follows:
[tex]\fbox{\begin\\\ \math y=mx+c\\\end{minispace}}[/tex]
In the above equation [tex]m[/tex] is the slope and [tex]c[/tex] is the [tex]y[/tex]-intercept.
From equation (2) and slope intercept form of the line it is observed that value of [tex]m[/tex] is [tex]\frac{-5}{2}[/tex].
This implies that the slope of the first line is [tex]\frac{-5}{2}[/tex].
Since, the first line and the second line are parallel to each other so their slope must be equal because two lines are said to be parallel if they have equal slope.
Therefore, the slope of the second line is [tex]\frac{-5}{2}[/tex].
For the second line it is known that the slope for the line is [tex]\frac{-5}{2}[/tex] and the line passes through the points [tex](-2,4)[/tex].
The general equation of the point slope form of a line is as follows:
[tex]\fbox{\begin\\\ \math (y-y_{1})=m(x-x_{1})\\\end{minispace}}[/tex] (3)
In the above equation [tex]m[/tex] is the slope and [tex](x_{1},y_{1})[/tex] is the point through the line passes.
To obtain the equation of the second line substitute [tex]\frac{-5}{2}[/tex] for [tex]m[/tex] and [tex](-2,4)[/tex] for [tex](x_{1},y_{1})[/tex] in equation (3).
[tex]\begin{aligned}(y-4)&=\frac{-5}{2}(x+2)\\y-4&=\frac{-5x}{2}-5\\y&=\frac{-5x}{2}-1\end{aligned}[/tex]
Therefore, the equation of the second line is [tex]y=\frac{-5x}{2}-1[/tex].
The options given are as follows:
Option1: [tex]y=\frac{-5x}{2}-1[/tex]
Option2: [tex]y=\frac{-5x}{2}+5[/tex]
Option3: [tex]y=\frac{2x}{5}-1[/tex]
Option4: [tex]y=\frac{2x}{5}+5[/tex]
So, as per the calculation made above option 1 is the correct option.
Figure 1 (attached in the end) shows that the line [tex]5x+2y=12[/tex] and [tex]y=\frac{-5x}{2}-1[/tex] are the parallel lines.
Thus, the equation of the line which is parallel to the line [tex]5x+2y=12[/tex] and passes through the point [tex](-2,4)[/tex] is [tex]y=\frac{-5x}{2}-1[/tex].
Learn more:
1. A problem to complete the square of quadratic function https://brainly.com/question/12992613
2. A problem to determine the slope intercept form of a line https://brainly.com/question/1473992
3. Inverse function https://brainly.com/question/1632445
Answer details
Grade: High school
Subject: Mathematics
Chapter: Lines
Keywords: Equation, line, slope, intercept, y-intercept, slope intercept form, point-slope form, parallel lines, equal slopes, graph, curve, equation of a line, curve, (-2,4), 5x+2y=12, y=(-5/2)x-1.
