In this article, you will learn about the concept of the point-slope form of a line. From the concept of finding the slope of a line when two points are given, the point-slope form is derived. When we can find the equation of the straight line which is inclined at an x-axis angle and passes through a given point, we will use the point-slope form of a line.

On this page, we will discuss the 10th Grade Math concept of the Point-slope form of a line. We will try to make you understand the point-slope form, formulas, how to derive the point-slope form formula, solved examples, and so on. This point-slope formula is only used when we know the slope of the line and a point on the line.

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### Point-Slope Form of a Line Definition

The point-slope form is defined as when we find the equation of a line with one point and slope which is inclined at an angle to the positive direction of the x-axis in the anti-clockwise direction and passes through a given point by an equation of a line in point-slope form. This is represented as using slope and point on the line of the straight line. The point-slope form of a line figure is as shown below,

### Point Slope Form Formula

In an equation of a line, the slope of a line is ‘m’ and which passes through a point (x_{1}, y_{1}). For finding the equation of a straight line we can express it in different forms such as slope-intercept form, intercept form, normal form, point-slope form, etc. The equation of a straight line using the point-slope form formula is,

(y-y_{1}) = m(x-x_{1})

- Where (x, y) is a random point on the line.
- m is the slope of the line.
- (x
_{1},y_{1}) is a fixed point on the line.

### Point-Slope Form Formula Derivation

Now, we will derive the formula of the equation of a straight line using the point-slope form. Consider that the slope of a line is ‘m’ and the points on the line are (x_{1},y_{1}), (x,y) be random points on the line whose coordinates are unknown. The point-slope form of a line formula is,

Point-slope form of a line is (y-y_{1}) = m(x-x_{1}).

We know the formula for the slope of a line.

So, the Slope of a line = (Difference in y-coordinates)/(Difference in x-coordinates)

i.e., m = (y-y_{1})/(x-x_{1}).

Now Multiply both sides by (x-x_{1}). Then we get

m(x-x_{1}) = ((y-y_{1})/(x-x_{1})) x (x-x_{1}).

After multiplying, this can be

m(x-x_{1})= (y-y_{1})

Finally, rewrite the equation such that the y-variables are on the left side to get to the desired form.

(y-y_{1}) = m(x-x_{1})

Hence, proved the point-slope form formula.

### How do you find the Point-Slope Form of a Line?

The following are the steps to find the equation of the straight line using the point-slope form of a line. The steps are:

**Step 1:** First, note down the slope of a line ‘m’, and the points that are lies on the line are coordinate points (x_{1},y_{1}).

**Step 2:** Now, Substitute the given values in the point-slope formula. The formula is (y-y_{1}) = m(x-x_{1})

**Step 3:** Finally, Simplify the equation to get the equation of a line in the standard form.

Read More:

- Coordinate Geometry Graph
- Graph of Standard Linear Relations Between x, y
- Slope of the Graph of y = mx + c

### Point-Slope Form Example Problems with Solutions

**Problem 1:** Find the equation of a line that passes through (3, -4) and with a slope is -5.

**Solution:
**Given that,

The points (x

_{1},y

_{1}) are (3,-4).

Next, the slope of a line is m =-5.

We know the formula of the equation of a line using a point-slope form is,

(y-y

_{1}) = m(x-x

_{1})

After substituting the given values in the above formula. It will be,

y-(−4) = (−5)(x − 3)

i.e., y+4 = -5x +15

5x +y+4 -15 =0

5x+y-11 =0

Hence, the equation of the line is 5x+y – 11 =0.

**Problem 2: **Find the equation of a line that passes through the points (2, –2) and (3, 5) in point-slope form.

**Solution:
**In the given question,

The points (x

_{1 },y

_{1}) is (2, -2).

The points of (x

_{2},y

_{2}) are (3, 5).

The formula for the slope of the line passes through the points is.

m = (y

_{2}– y

_{1})/ (x

_{2}– x

_{1})

After substituting the given values in the above formula. Then we get,

m = (5-(-2))/(3-2)

m= 7/1

So, the slope of the line m is 7.

The equation of a line passing through the point (2, -2) with slope 7.

The formula for equation of a line using point-slope form is,

(y-y

_{1}) = m(x-x

_{1})

y -(-2) = 7(x – 2)

y + 2 = 7x – 14

-7x + y + 2 + 14 = 0

-7x + y + 16= 0

Similarly, the equation of a line passing through the point (3, 5) with slope of a line 7.

Substitute these value in a point slope from formula. Then we get,

y – 5 = 7(x – 3)

y – 5 = 7x – 21

-7x + y – 5 + 21= 0

-7x + y + 16 = 0

Therefore, the equation of the line in point slope form is -7x + y + 16 = 0.

**Problem 3:** A straight line that passes through the point (4, -6) and the positive direction of the x-axis gives an angle of 135 °. Find the equation for a straight line?

**Solution:**

As given in the question,

The points (x_{1 },y_{1}) are (4,-6)

An angle of 135 ° with a positive direction of the x-axis line.

The slope of the line m is,

m= tan 135 ° = tan (90 ° + 45 °) = – cot 45 ° = -1.

The line that is needed to pass through the point (4, -6).

We know the equation of a line using the point-slope formula.

The formula is (y-y_{1}) = m(x-x_{1})

Substitute the values in the above formula. We get,

y – (-6) = -1 (x -4)

y + 6 = -x + 4

x + y + 6- 4 = 0

x + y + 2 = 0

Thus, the equation of a straight line is x + y + 2 = 0.

**Problem 4:** Find the slope of the line when the equation of the line is 2x – 3y + 1.

**Solution:
**Given that, the equation of a line is 2x-3y+1

Now, we will find the value of the slope of a line.

The given equation of a line is in the form of ax + by + c is -a/b

So, the value of a is 2 and the value of the b is -3.

then the value of the m is -(2/-3)

Hence, the slope of a line is 2/3.

**Problem 5: **Find the equation of the line which passes through two points (3,6) and (4,8).

**Solution: **

In the given question,

The points (x_{1 },y_{1}) is (3, 6).

The points of (x_{2}, y_{2}) are (4, 8).

The two points are given. So, the two points slope form can be

y-y_{1} = (y_{2}-y_{1})/(x_{2}-x_{1}) * (x-x_{1})

Substitute the values in the above formula as x_{1} = 3, y_{1} = 6, x_{2} = 4, y_{2} = 8

then the equation is y -6 = (8-6 )/(4-3) * (x-3)

y-6 = 2*(x-3)

y -6 = 2x-6

2x-y =0

Hence, the equation of the line is 2x-y = 0

### FAQ’s on Point-Slope Form of a Line

**1. How do you find the slope for two points?**

The two points (x_{1}, y_{1}) and (x_{2}, y_{2}) be given, then the slope of the line passing through these points is: m = (y2 – y1)/(x2 – x1)

**2. When can we use the point-slope of a line formula?**

The formula of the point-slope form of an equation is used when we need to find the equation of a straight line given a point on it and the slope of the line.

**3. How do you Change Point-Slope Form into the Slope-Intercept Form?**

The formula for the point-slope form of a line is (y-y_{1}) = m(x-x_{1}). We will solve this equation for y which gives an equation in the form is y = mx + b. This is called the slope-intercept form.

**4. What are the Applications of the Point-Slope Formula?**

The applications of the point-slope of a line formula are,

- We can find the equation of a line with the given slope and a point on it.
- It can be used to Graph the line by using the slope and one point on the line.
- One can find the slope of the line right away from the equation of the line.