Give the equation of a line with a known slope and point

The slope-intercept form has the advantage of being simple to remember and use, however, it has one major disadvantage: we must know the y-intercept in order to use it! Generally we do not know the y-intercept, we only know one or more points (that are not the y-intercept). In these cases we can’t use the slope intercept equation, so we will use a different more flexible formula. If we let the slope of an equation be m, and a specific point on the line be (x_1,y_1), and any other point on the line be (x, y). We can use the slope formula to make a second equation.

Also Read : How to Give the equation of a line with a known slope and y-intercept

Example 1:

Also Read:

Find the equation on the line (x_1,y_1) and any other point on the line be (x,y)

Solution:

Recall slope formula

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

Plug in value

\frac{y-y_1}{x-x_1}=m

Multiply both sides by (x-x_1)

So, the solution is

y-y_1=m(x-x_1)

If we know the slope, m of an equation and any point on the line (x_1,y_1) we can easily plug these values into the equation above which will be called the point- slope formula.

Point Slope Formula:  y-y_1=m(x-x_1)

Example 2:

Write the equation of the line through the point (3, -4) with a slope of \frac{3}{5}

Solution:

Plug values into point – slope formula

y-y_1=m(x-x_1)

y-(-4)=\frac{3}{5}(x-3)

Simplify signs, and the solution is

y+4=\frac{3}{5}(x-3)

Often, we will prefer final answers be written in slope intercept form. If the directions ask for the answer in slope-intercept form we will simply distribute the slope, then solve for y.

Also Read :  Identify the equation of a line given a parallel or perpendicular line

Example 3:

Write the equation of the line through the point ( – 6, 2) with a slope of \frac{-2}{3} in slope-intercept form.

Solution :

Plug values into point -slope formula

y-y_1=m(x-x_1)

y-2=-\frac{2}{3}(x-(-6))

y-2=-\frac{2}{3}(x+6)

Solve for y

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

Add +2 to both sides

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

And the solution is:

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

An important thing to observe about the point slope formula is that the operation between the x’s and y’s is subtraction. This means when you simplify the signs you will have the opposite of the numbers in the point. We need to be very careful with signs as we use the point-slope formula.

Also Read : How to Find the slope of a line given a graph or two points

In order to find the equation of a line we will always need to know the slope. If we don’t know the slope to begin with we will have to do some work to find it first before we can get an equation.

Example 4:

Find the equation of the line through the points ( -2, 5) and (4, -3).

Solution:

First we must find the slope

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

Plug values in slope formula and evaluate

m=\frac{-3-5}{4-(-2)}=\frac{-8}{6}=-\frac{4}{3}

With slope and either point, use point -slope formula

y-y_1=m(x-x_1)

y-5=-\frac{4}{3}(x-(-2))

Simplify signs and the solutions is

y-5=-\frac{4}{3}(x+2)

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