Question

Through (2,-4) parallel to y=3x+2

Answer 1

Slope-intercept form: is y=3x-10.

Point-slope form: y+4=3(x-2).

Standard form: 3x-y=10.

Explanation:

Slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.

A line parallel to y=3x+2 will be of the form y=3x+b where b is not 2.

To find b, we will use that our line goes through (2,-4).

-4=3(2)+b

-4=6+b

Subtract 6 on both sides:

-4-6=b

-10=b

The the line we are looking for in slope-intercept form is y=3x-10.

Standard form is ax+by=c.

y=3x-10

Add 10 on both sides:

10+y=3x

Subtract y on both sides:

10=3x-y

So the equation in standard form is 3x-y=10.

Point-slope form is
`y-y_1=m(x-x_1)`.

`y-(-4)=3(x-2)`

`y+4=3(x-2)`

Answer 2

y = 3x - 10

Explanation:

`\text{The slope-intercept form of an equation of a line:}\\\\y=mx+b\\\\m-\text{slope}\\b-\text{y-intercept}\\\\\text{Let}\ k:y=m_1x+b_1,\ l:y=m_2x+b_2\\\\\text{then}\\\\l\ \parallel\ k\iff m_1=m_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)`

`\text{Conclusion:}\\\\\text{Parallel lines have the same slope.}`

`\text{We have}\ y=3x+2\to m=3`

`\text{and the point}\ (2,\ -4)\to x=2,\ y=-4.`

`\text{Substitute to the equation of a line:}\\\\-4=(3)(2)+b\\\\-4=6+b\qquad\text{subtract 6 from both sides}\\\\-10=b\to b=-10`

`\text{Finally:}\\\\y=3x-10`