I really never got this type of problem, I never really got graphs and slopes

## What is the equation of the line that passes through the points (2, 5) and (1, 2)?

# 4 Answers

Hi Rishma... You need to use the "point-slope" formula to solve this question. you already have 2 points...so now you need the slope...

Like Craig said, the first step is to find the slope of the line...this is "rise over run".

so ask yourself, "How far did i rise from the first point to the second point(the 'y' values)?" answer....from 2 to 5 takes 3 positive steps so...3 is the "rise" .

now, do the same with the "run".

ask yourself, "How far did i run from the first point to the second point?" answer....from 1 to 2 takes 1 positive step so...1 is the "run" . This gives a slope of 3/1.

Next, you have to put this slope and one of the points into that "point-slope" formula.

y-y_{1 }=m(x-x_{1 }) ----> we will use the point (1,2) = (x_{1},y_{1}) here

y-2=3(x-1)=3x-3

add 2 to both sides and you get your equation...

y = 3x-1

**Solution**: **
y-5=3(x-2)** or **y-2=3(x-1)** in* point slope form*;

**y=3x-1**in

**intercept slope form**To understand why, please read the following step by step solution.

**STEP 1**: **
Read**, **understand** the situation within, **identify** and pull out
**important** **information**.

• Consider the* point slope form* and the

*.*

**intercept slope form**equations• Both equations need the * slope “m”*, and the

*on the line.*

**coordinates of a point**• Calculate the slope “m” using the coordinates of the two given points (2,5) and (1,2).

• **Slope** “m”= **RISE/RUN**; be congruent when going from one point to another point of the line. You have to get the same result.

RISE= Change in y: (y2-y1); RUN= Change in x: (x2-x1). You’re going from P1 to P2 (ending point)

RISE= Change in y: (y1-y2); RUN= Change in x: (x1-x2). You’re going from P2 to P1 (ending point)

• For the * point slope form:* use the coordinates of one of the two given points. If you chose the other point (or any other point of the line) you’ll get an equivalent point slope equation for the same line!

• For the * intercept slope form: *we would need the y-intercept, but we don't have the point where the line crosses the y-axis. We can get the intercept slope form from the point slope form by isolating "y". Remember: the slope intercept
form is a particular case of the point slope form, because the point used is the y-intercept point (0,b).

**STEP 2**: **
Translate** **keywords** to their mathematical symbols:

• Point slope form equation: **y-y1=m(x-x1)** or **y-y2=m(x-x2)** It does not matter!

• Intercept slope form equation: **y=mx+b** b is the y-intercept of the intercept point (0,y)

• Slope of the line: **m = (y2-y1) / (x2-x1)** or **m = (y1-y2) / (x1-x2)** It doesn't matter, just be congruent!

**STEP 3**: **
Set up** and **solve **the equation or problem:

• **Slope “m”**:

m= (2-5) / (1-2) or m = (5-2) / (2-1)

m = -3 / (-1) or m = 3 / 1

**m = 3** or m = 3

• **Point slope form equation**:

y-y1 = m (x-x1) or y-y2 = m (x-x2)

**y-5 = 3 (x-2)** *using (2,5)* or **y-2 = 3 (x-1)**
*using (1,2)*

These are two point slope form equations for the line passing through (2,5) and (1,2)

• **Intercept slope form equation**: (Starting from the point slope form equation)

y-5 = 3 (x-2) or y-2 = 3 (x-1) apply the Distributive Property of Multiplication

y-5 = 3x-6 or y-2 = 3x-3

+5 +5 or +2 +2 to isolate "y"

** y = 3x-1** or **y = 3x-1** This is the intercept slope form equation for the line through (2,5) and (1,2)

**STEP 4**: **
Check** the solution:

• For the **point slope form** equation:

y-5 = 3 (x-2) or y-2 = 3 (x-1)

5-5 = 3 (2-2) or 5-2 = 3 (2-1) substituting the coordinates of the given point (2,5)

0 = 3 (0) or 3 = 3 (1) Applying the Distributive Property of Multiplication

0 = 0 or 3 = 3 Both are identities, so the two equations represent the point slope form equations of the same line through the points (2,5), (1,2).

• For the **intercept slope form** equation:

y = 3x-1 or y = 3x-1

5 = 3(2)-1 or 2 = 3(1)-1 substituting the coordinates (2,5) and (1,2) respectively

5 = 6-1 or 2 = 3-1

5 = 5 or 2 = 2 Both are identities, so the equation is the intercept slope form equation of the line through the points (2,5), (1,2).

**STEP 5:** **Curiosities**

- There are
representing the same line! In fact, meanwhile “m” remains always the same (it’s the same line!), by selecting a different point of the line, we are just changing the coordinates (x1,y1), and so we are getting another equation. Even though these equations look different they are equivalent.**infinite equivalent point slope form equations**

- The
in representing a line because it has**intercept slope form equation is just unique**!**only one y-intercept point**

- Do you want to graph this line? Just plot the two given points (2,5) and (1,2) in a (x,y) Cartesian plane. Then draw a line through these two points! REMEMBER: there is only one line passing through two points.
**Two points define a unique line!**

*Getting the slope intercept form from the point slope form:*

* y-y1 = m(x-x1) Substitute the y-intercept point coordinates (0.b) *

* y-b = m(x-0) *

* +b +b *

* y = mx - m(0) + b *

* y = mx + b tutto bene!*

* *

We know that the general equation of a line passing through two points (x1, y1) and ( x2, y2) is,

(x-x1)/(x1-x2) = (y-y1)/(y1-y2)

substituting the values of x1, y1, x2, y2,

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

or, x-2 = (y-5)/3

or, 3x-6 = y-5

or 3x-y=1 **So the equation of the line is, 3x-y=1**

Take these problems step by step. subtract y1 from y2. subtract x1 from x2. 2-1=1 and 5-2=3. divide 1 by 3 or 1/3 or .3333 is the slope.