Point Slope Form

Point-slope form is used to write the equation of a line when you know its slope and any point on the line. Substitute the slope and point into $y-y_1=m(x-x_1)$, then simplify when slope-intercept form is needed. This form is also useful for writing equations from intercepts, two points, parallel lines, perpendicular lines, and tangent-line information.

Lesson

Notes

Notes for Point Slope Form

Practice Problems

$\textbf{1)}$ Find the equation of the line with slope $3$ that goes through the point $(6,4)$.
Show Point-Slope Form
Point-slope form is $y-4=3(x-6)$.
Show Slope-Intercept Form
Slope-intercept form is $y=3x-14$.
Show Work

$y-y_1=m(x-x_1)$

$y-4=3(x-6)$

$y-4=3x-18$

$y=3x-14$

$\textbf{2)}$ Find the equation of the line with slope $-2$ that goes through the point $(-1,5)$.
Show Point-Slope Form
Point-slope form is $y-5=-2(x+1)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-2x+3$.
Show Work

$y-y_1=m(x-x_1)$

$y-5=-2(x-(-1))$

$y-5=-2(x+1)$

$y-5=-2x-2$

$y=-2x+3$

$\textbf{3)}$ Find the equation of the line with slope $\displaystyle\frac{1}{3}$ that goes through the point $(3,8)$.
Show Point-Slope Form
Point-slope form is $y-8=\frac{1}{3}(x-3)$.
Show Slope-Intercept Form
Slope-intercept form is $y=\frac{1}{3}x+7$.
Show Work

$y-y_1=m(x-x_1)$

$y-8=\frac{1}{3}(x-3)$

$y-8=\frac{1}{3}x-1$

$y=\frac{1}{3}x+7$

$\textbf{4)}$ Find the equation of the line with slope $1$ that goes through the point $(-6,-4)$.
Show Point-Slope Form
Point-slope form is $y+4=1(x+6)$.
Show Slope-Intercept Form
Slope-intercept form is $y=x+2$.
Show Work

$y-y_1=m(x-x_1)$

$y-(-4)=1(x-(-6))$

$y+4=x+6$

$y=x+2$

$\textbf{5)}$ Find the equation of the line with slope $\frac{1}{2}$ and x-intercept $4$.
Show Hint
An x-intercept of $4$ means $(4,0)$ is a point on the line.
Show Point-Slope Form
Point-slope form is $y-0=\frac{1}{2}(x-4)$.
Show Slope-Intercept Form
Slope-intercept form is $y=\frac{1}{2}x-2$.
Show Work

$\text{x-intercept }4\longrightarrow(4,0)$

$y-y_1=m(x-x_1)$

$y-0=\frac{1}{2}(x-4)$

$y=\frac{1}{2}x-2$

$\textbf{6)}$ Find the equation of the line with slope $-5$ and y-intercept $-2$.
Show Hint
A y-intercept of $-2$ means $(0,-2)$ is a point on the line.
Show Point-Slope Form
Point-slope form is $y+2=-5(x-0)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-5x-2$.
Show Work

$\text{y-intercept }-2\longrightarrow(0,-2)$

$y-y_1=m(x-x_1)$

$y-(-2)=-5(x-0)$

$y+2=-5x$

$y=-5x-2$

$\textbf{7)}$ Find the equation of the line with slope $-4$ that goes through the point $(2,-3)$.
Show Point-Slope Form
Point-slope form is $y+3=-4(x-2)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-4x+5$.
Show Work

$y-y_1=m(x-x_1)$

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

$y+3=-4x+8$

$y=-4x+5$

$\textbf{8)}$ Find the equation of the line with slope $\frac{3}{4}$ that goes through the point $(-4,1)$.
Show Point-Slope Form
Point-slope form is $y-1=\frac{3}{4}(x+4)$.
Show Slope-Intercept Form
Slope-intercept form is $y=\frac{3}{4}x+4$.
Show Work

$y-y_1=m(x-x_1)$

$y-1=\frac{3}{4}(x-(-4))$

$y-1=\frac{3}{4}(x+4)$

$y-1=\frac{3}{4}x+3$

$y=\frac{3}{4}x+4$

$\textbf{9)}$ Find the equation of the line that goes through the points $(1,2)$ and $(5,10)$.
Show Point-Slope Form
Point-slope form is $y-2=2(x-1)$.
Show Slope-Intercept Form
Slope-intercept form is $y=2x$.
Show Work

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{10-2}{5-1}$

$m=\frac{8}{4}=2$

$y-2=2(x-1)$

$y-2=2x-2$

$y=2x$

$\textbf{10)}$ Find the equation of the line that goes through the points $(-2,7)$ and $(4,-5)$.
Show Point-Slope Form
Point-slope form is $y-7=-2(x+2)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-2x+3$.
Show Work

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{-5-7}{4-(-2)}$

$m=\frac{-12}{6}=-2$

$y-7=-2(x-(-2))$

$y-7=-2(x+2)$

$y-7=-2x-4$

$y=-2x+3$

$\textbf{11)}$ Find the equation of the line that goes through the points $(3,-1)$ and $(9,2)$.
Show Point-Slope Form
Point-slope form is $y+1=\frac{1}{2}(x-3)$.
Show Slope-Intercept Form
Slope-intercept form is $y=\frac{1}{2}x-\frac{5}{2}$.
Show Work

$m=\frac{y_2-y_1}{x_2-x_1}$

$m=\frac{2-(-1)}{9-3}$

$m=\frac{3}{6}=\frac{1}{2}$

$y-(-1)=\frac{1}{2}(x-3)$

$y+1=\frac{1}{2}x-\frac{3}{2}$

$y=\frac{1}{2}x-\frac{5}{2}$

$\textbf{12)}$ Find the equation of the line parallel to $y=4x-7$ that goes through the point $(-3,5)$.
Show Point-Slope Form
Point-slope form is $y-5=4(x+3)$.
Show Slope-Intercept Form
Slope-intercept form is $y=4x+17$.
Show Work

$\text{Parallel lines have the same slope.}$

$m=4$

$y-5=4(x-(-3))$

$y-5=4(x+3)$

$y-5=4x+12$

$y=4x+17$

$\textbf{13)}$ Find the equation of the line parallel to $2x+5y=15$ that goes through the point $(5,-2)$.
Show Point-Slope Form
Point-slope form is $y+2=-\frac{2}{5}(x-5)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-\frac{2}{5}x$.
Show Work

$2x+5y=15$

$5y=-2x+15$

$y=-\frac{2}{5}x+3$

$\text{The parallel slope is }m=-\frac{2}{5}.$

$y-(-2)=-\frac{2}{5}(x-5)$

$y+2=-\frac{2}{5}x+2$

$y=-\frac{2}{5}x$

$\textbf{14)}$ Find the equation of the line perpendicular to $y=2x+9$ that goes through the point $(4,3)$.
Show Point-Slope Form
Point-slope form is $y-3=-\frac{1}{2}(x-4)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-\frac{1}{2}x+5$.
Show Work

$\text{The original slope is }2.$

$\text{The perpendicular slope is }-\frac{1}{2}.$

$y-3=-\frac{1}{2}(x-4)$

$y-3=-\frac{1}{2}x+2$

$y=-\frac{1}{2}x+5$

$\textbf{15)}$ Find the equation of the line perpendicular to $3x-4y=12$ that goes through the point $(-6,1)$.
Show Point-Slope Form
Point-slope form is $y-1=-\frac{4}{3}(x+6)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-\frac{4}{3}x-7$.
Show Work

$3x-4y=12$

$-4y=-3x+12$

$y=\frac{3}{4}x-3$

$\text{The original slope is }\frac{3}{4}.$

$\text{The perpendicular slope is }-\frac{4}{3}.$

$y-1=-\frac{4}{3}(x-(-6))$

$y-1=-\frac{4}{3}(x+6)$

$y-1=-\frac{4}{3}x-8$

$y=-\frac{4}{3}x-7$

$\textbf{16)}$ A taxi charges a fixed starting fee plus $\$2.50$ per mile. A $6$-mile trip costs $\$19$. Write an equation for the total cost $y$ of a trip of $x$ miles.
Show Point-Slope Form
Point-slope form is $y-19=2.5(x-6)$.
Show Slope-Intercept Form
Slope-intercept form is $y=2.5x+4$.
Show Work

$\text{The cost per mile is the slope, so }m=2.5.$

$\text{A 6-mile trip costing $19 gives the point }(6,19).$

$y-19=2.5(x-6)$

$y-19=2.5x-15$

$y=2.5x+4$

Challenge Problems

$\textbf{17)}$ Find the equation of the perpendicular bisector of the segment with endpoints $(-2,1)$ and $(4,5)$.
Show Point-Slope Form
Point-slope form is $y-3=-\frac{3}{2}(x-1)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-\frac{3}{2}x+\frac{9}{2}$.
Show Work

$m=\frac{5-1}{4-(-2)}$

$m=\frac{4}{6}=\frac{2}{3}$

$\text{The perpendicular slope is }-\frac{3}{2}.$

$\text{Midpoint}=\left(\frac{-2+4}{2},\frac{1+5}{2}\right)$

$\text{Midpoint}=(1,3)$

$y-3=-\frac{3}{2}(x-1)$

$y-3=-\frac{3}{2}x+\frac{3}{2}$

$y=-\frac{3}{2}x+\frac{9}{2}$

$\textbf{18)}$ Find the equation of the line that passes through the intersection of $y=2x-1$ and $y=-x+8$ and has slope $-4$.
Show Point-Slope Form
Point-slope form is $y-5=-4(x-3)$.
Show Slope-Intercept Form
Slope-intercept form is $y=-4x+17$.
Show Work

$2x-1=-x+8$

$3x=9$

$x=3$

$y=2(3)-1=5$

$\text{The lines intersect at }(3,5).$

$y-5=-4(x-3)$

$y-5=-4x+12$

$y=-4x+17$

$\textbf{19)}$ A line passes through $(k,7)$ and $(5,-1)$ and has slope $-2$. Find $k$, then write the equation of the line.
Show Answer
$k=1$, point-slope form is $y-7=-2(x-1)$, and slope-intercept form is $y=-2x+9$.
Show Work

$-2=\frac{-1-7}{5-k}$

$-2=\frac{-8}{5-k}$

$-2(5-k)=-8$

$-10+2k=-8$

$2k=2$

$k=1$

$y-7=-2(x-1)$

$y-7=-2x+2$

$y=-2x+9$

$\textbf{20)}$ The graph of a linear function passes through $(a,4)$ and $(a+6,-5)$. Find its slope and write the equation of the line when $a=2$.
Show Answer
The slope is $-\frac{3}{2}$, point-slope form is $y-4=-\frac{3}{2}(x-2)$, and slope-intercept form is $y=-\frac{3}{2}x+7$.
Show Work

$m=\frac{-5-4}{(a+6)-a}$

$m=\frac{-9}{6}$

$m=-\frac{3}{2}$

$a=2\longrightarrow(a,4)=(2,4)$

$y-4=-\frac{3}{2}(x-2)$

$y-4=-\frac{3}{2}x+3$

$y=-\frac{3}{2}x+7$

$\bullet\text{ Graphing Linear Equations}$
$\,\,\,\,\,\,\,\,2x-3y=6$

$\bullet\text{ Slope Formula}$
$\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}$

$\bullet\text{ Net Change}$
$\,\,\,\,\,\,\,\,y_2-y_1$

$\bullet\text{ Slope-Intercept Form}$
$\,\,\,\,\,\,\,\,y=mx+b$

$\bullet\text{ Parallel and Perpendicular Slope}$
$\,\,\,\,\,\,\,\,m_1=m_2,\,\,\,m_1=-\frac{1}{m_2}$

$\bullet\text{ Distance Between a Point and a Line}$
$\,\,\,\,\,\,\,\,(3,4)\text{ and }y=\frac{3}{4}x-2$

$\bullet\text{ Finding x- and y-intercepts}$
$\,\,\,\,\,\,\,\,y=2x+4$

ANDYMATH

Lesson

Notes

Practice Problems

\(\textbf{1)}\) Find the equation of the line with slope \(3\) that goes through the point \((6,4)\).
Show Point-Slope Form
Point-slope form is \(y-4=3(x-6)\).
Show Slope-Intercept Form
Slope-intercept form is \(y=3x-14\).
Show Work

\(y-y_1=m(x-x_1)\)

\(y-4=3(x-6)\)

\(y-4=3x-18\)

\(y=3x-14\)

\(\textbf{4)}\) Find the equation of the line with slope \(1\) that goes through the point \((-6,-4)\).
Show Point-Slope Form
Point-slope form is \(y+4=1(x+6)\).
Show Slope-Intercept Form
Slope-intercept form is \(y=x+2\).
Show Work

\(y-y_1=m(x-x_1)\)

\(y-(-4)=1(x-(-6))\)

\(y+4=x+6\)

\(y=x+2\)

Challenge Problems

Related Pages

\(\bullet\text{ Graphing Linear Equations}\)
\(\,\,\,\,\,\,\,\,2x-3y=6\)

\(\bullet\text{ Slope Formula}\)
\(\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}\)

\(\bullet\text{ Net Change}\)
\(\,\,\,\,\,\,\,\,y_2-y_1\)

\(\bullet\text{ Slope-Intercept Form}\)
\(\,\,\,\,\,\,\,\,y=mx+b\)

\(\bullet\text{ Parallel and Perpendicular Slope}\)
\(\,\,\,\,\,\,\,\,m_1=m_2,\,\,\,m_1=-\frac{1}{m_2}\)

\(\bullet\text{ Distance Between a Point and a Line}\)
\(\,\,\,\,\,\,\,\,(3,4)\text{ and }y=\frac{3}{4}x-2\)

\(\bullet\text{ Finding x- and y-intercepts}\)
\(\,\,\,\,\,\,\,\,y=2x+4\)

ANDYMATH

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ANDYMATH

Lesson

Notes

Practice Problems

\(\textbf{1)}\) Find the equation of the line with slope \(3\) that goes through the point \((6,4)\).Show Point-Slope FormPoint-slope form is \(y-4=3(x-6)\).Show Slope-Intercept FormSlope-intercept form is \(y=3x-14\).Show Work \(y-y_1=m(x-x_1)\) \(y-4=3(x-6)\) \(y-4=3x-18\) \(y=3x-14\)

\(\textbf{4)}\) Find the equation of the line with slope \(1\) that goes through the point \((-6,-4)\).Show Point-Slope FormPoint-slope form is \(y+4=1(x+6)\).Show Slope-Intercept FormSlope-intercept form is \(y=x+2\).Show Work \(y-y_1=m(x-x_1)\) \(y-(-4)=1(x-(-6))\) \(y+4=x+6\) \(y=x+2\)

\(\textbf{7)}\) Find the equation of the line with slope \(-4\) that goes through the point \((2,-3)\).Show Point-Slope FormPoint-slope form is \(y+3=-4(x-2)\).Show Slope-Intercept FormSlope-intercept form is \(y=-4x+5\).Show Work \(y-y_1=m(x-x_1)\) \(y-(-3)=-4(x-2)\) \(y+3=-4x+8\) \(y=-4x+5\)

Challenge Problems

Related Pages

\(\bullet\text{ Graphing Linear Equations}\) \(\,\,\,\,\,\,\,\,2x-3y=6\)

\(\bullet\text{ Slope Formula}\) \(\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}\)

\(\bullet\text{ Net Change}\) \(\,\,\,\,\,\,\,\,y_2-y_1\)

\(\bullet\text{ Slope-Intercept Form}\) \(\,\,\,\,\,\,\,\,y=mx+b\)

\(\bullet\text{ Parallel and Perpendicular Slope}\) \(\,\,\,\,\,\,\,\,m_1=m_2,\,\,\,m_1=-\frac{1}{m_2}\)

\(\bullet\text{ Distance Between a Point and a Line}\) \(\,\,\,\,\,\,\,\,(3,4)\text{ and }y=\frac{3}{4}x-2\)

\(\bullet\text{ Finding x- and y-intercepts}\) \(\,\,\,\,\,\,\,\,y=2x+4\)

\(\textbf{1)}\) Find the equation of the line with slope \(3\) that goes through the point \((6,4)\).
Show Point-Slope Form
Point-slope form is \(y-4=3(x-6)\).
Show Slope-Intercept Form
Slope-intercept form is \(y=3x-14\).
Show Work

\(y-y_1=m(x-x_1)\)

\(y-4=3(x-6)\)

\(y-4=3x-18\)

\(y=3x-14\)

\(\textbf{4)}\) Find the equation of the line with slope \(1\) that goes through the point \((-6,-4)\).
Show Point-Slope Form
Point-slope form is \(y+4=1(x+6)\).
Show Slope-Intercept Form
Slope-intercept form is \(y=x+2\).
Show Work

\(y-y_1=m(x-x_1)\)

\(y-(-4)=1(x-(-6))\)

\(y+4=x+6\)

\(y=x+2\)

\(\bullet\text{ Graphing Linear Equations}\)
\(\,\,\,\,\,\,\,\,2x-3y=6\)

\(\bullet\text{ Slope Formula}\)
\(\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}\)

\(\bullet\text{ Net Change}\)
\(\,\,\,\,\,\,\,\,y_2-y_1\)

\(\bullet\text{ Slope-Intercept Form}\)
\(\,\,\,\,\,\,\,\,y=mx+b\)

\(\bullet\text{ Parallel and Perpendicular Slope}\)
\(\,\,\,\,\,\,\,\,m_1=m_2,\,\,\,m_1=-\frac{1}{m_2}\)

\(\bullet\text{ Distance Between a Point and a Line}\)
\(\,\,\,\,\,\,\,\,(3,4)\text{ and }y=\frac{3}{4}x-2\)

\(\bullet\text{ Finding x- and y-intercepts}\)
\(\,\,\,\,\,\,\,\,y=2x+4\)