Question

Solve

5x+9=3x+25

Write an equation in slope intercept form for a line that goes through the point (-9,4) and is perpendicular to the line
y=3x-2

Write an equation in slope intercept form for a line that goes through the point (2, -8) and has a slope of 5

Solve 5x=3y-w for y

Answer 1

`\large1.\ x=8\\\\2.\ y=-(1)/(3)x+7\\\\3.\ y=5x-18\\\\4.\ y=(5x+w)/(3)`

Explanation:

`1.\\5x+9=3x+25\qquad\text{subtract 9 from both sides}\\\\5x=3x+16\qquad\text{subtract 3x from both sides}\\\\2x=16\qquad\text{divide both sides by 2}\\\\\boxed{x=8}`

`2.\\\text{The slope-intercept form:}\\y=mx+b\\\\m-slope\\b-y-intercept\\\\\text{Let}\ k:y=m_1x+b_1,\ l:y=m_2x+b_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)\\\\\text{Given}\\\\y=3x-2\to m_1=3\\\\m_2=-(1)/(3)\\\\\text{Therefore we have}\ y=-(1)/(3)x+b.\\\\\text{Substitute the coordinates of the point (-9, 4) to the equation of a line:}\\\\4=-(1)/(3)(9)+b\\\\4=-3+b\qquad\text{add 3 to both sides}\\\\7=b\to b=7\\\\\text{Finally}\\\\y=-(1)/(3)x+7`

`3.\\y=mx+b\\\\\text{We have the slope}\ m=5\ \text{and the point}\ (2,\ -8).\\\\\text{Therefore}\ y=5x+b.\ \text{Substitute the coordinates of the point to the equation:}\\\\-8=5(2)+b\\\\-8=10+b\qquad\text{subtract 10 from both sides}\\\\-18=b\to b=-18\\\\\text{Finally}\\\\y=5x-18`

`4.\\3y-w=5x\qquad\text{add}\ w\ \text{to both sides}\\\\3y=5x+w\qquad\text{divide both sides by 3}\\\\y=(5x+w)/(3)`