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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The equation of a line can be written in slope-intercept form or standard form. However, when the slope and a point are given, point-slope form is another alternative.

$y−y_{1}=m(x−x_{1})$

As its name suggests, point-slope form requires a point on and the slope of the line.

To write the point-slope form for the line that passes through the points $(-1,5)and(1,1),$ find the slope, then use either point to write the equation.Find the slope

$m=x_{2}−x_{1}y_{2}−y_{1} $

$m=1−(-1)1−5 $

SubTermSubtract term

$m=1−(-1)-4 $

SubNeg$a−(-b)=a+b$

$m=2-4 $

CalcQuotCalculate quotient

$m=-2$

Choose one point on the line

Substitute values

Write the equation for the line in point-slope form.

Show Solution

To write the equation of the line in point-slope form, $y−y_{1}=m(x−x_{1}),$ we need the slope and a point.

Since we're given the line as a graph we can use the rise and run to find the slope. We'll find the rise and run between two arbitratily chosen points, the $y$-intercept, $(0,-2),$ and $(2,1).$

Here, the run is $2$ and the rise is $3,$ which gives a slope of $m=23 .$

Next, any point on the line can be used for $(x_{1},y_{1}).$ Let's use the same point as above, $(2,1).$

Lastly, substitute the found values of $m$ and $(x_{1},y_{1})$ into $y−y_{1}=m(x−x_{1}).$ Here, $m=23 $ and $(x_{1},y_{1})=(2,1)$ will be substituted. $y−1=23 (x−2).$

All points in the table lie on the same line.

Write the equation of the line in point-slope form.

Show Solution

To write an equation in the form $y−y_{1}=m(x−x_{1}),$ the slope of the line, $m,$ and any point on the line, $(x_{1},y_{1}),$ must be known. We'll begin by finding the slope between consecutive points. It can be seen that the difference between each $x$-value is $1.$

In the right column, we see that the difference between each value is $2.$

Therefore, $m=2.$ Next, we need any point on the line. We know that all points in the table are on the line, so we can choose any of them. Let's use $(-5,-1)$ to be $(x_{1},y_{1}.)$ Substituting this point and $m=2$ gives us the line.$y−y_{1}=m(x−x_{1})$

Substitute$m=2$

$y−y_{1}=2(x−x_{1})$

$y−(-1)=2(x−(-5))$

SubNeg$a−(-b)=a+b$

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

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