Find the equation of the line which passes through the point (1 2 3)

Slope-Intercept:

The following video will teach how to find the equation of a line, given any two points on that line.

Video Source (7:13 mins) | Transcript

Steps to find the equation of a line from two points:

1. Find the slope using the slope formula
   • \({\text{Slope}}={\text{m}}=\frac{\text{rise}}{\text{run}}=\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}\)
   • \(\text{Point 1 or P}_{1}=(\text{x}_{1}, \text{y}_{1})\)
   • \(\text{Point 2 or P}_{2}=(\text{x}_{2}, \text{y}_{2})\)
2. Use the slope and one of the points to solve for the y-intercept (b).
   • One of your points can replace the x and y, and the slope you just calculated replaces the m of your equation y = mx + b. Then b is the only variable left. Use the tools you know for solving for a variable to solve for b.
3. Once you know the value for m and the value for b, you can plug these into the slope-intercept form of a line (y = mx + b) to get the equation for the line.

Additional Resources

• Khan Academy: Slope-Intercept Equation from Two Points (06:41 mins, Transcript)
• Khan Academy: Slope-Intercept Form Problems (14:57 mins, Transcript)

Practice Problems

For each of the following problems, find the equation of the line that passes through the following two points:

1. \(\left ( -5,10 \right )\) and \(\left ( -3,4 \right )\)
2. \(\left ( -5,-26 \right )\) and \(\left ( -2,-8 \right )\)
3. \(\left ( -4,-22 \right )\) and \(\left ( -6,-34 \right )\)
4. \(\left ( 3,1 \right )\) and \(\left ( -6,-2 \right )\)
5. \(\left ( 4,-6 \right )\) and \(\left ( 6,3 \right )\)
6. \(\left ( 5,5 \right )\) and \(\left ( 3,2 \right )\)

The general equation of a straight line can be written as y = mx + c where m is the slope and c is the y-intercept.

Answer: The equation of a line passing through the points (-1, 3) and (2, -3) is 2x +y = + 1.

Let us proceed step by step to write an equation of a line.

Explanation:

Let us consider the given points (-1,3) and (2,-3).

As we know that the equation of a line passing through the points (x1, y1) and (x2 , y2) is given by y - y1 = m (x - x1).

Here, m is the slope given by the formula m = (y2 - y1) / (x2 - x1)

You can find the slope using Cuemath's Slope Calculator.

Hence on substituting the given points in the equation of a line, we get

y - 1 = m ( x - (-1) ) -------(1)

m =  (y2 - y1) / (x2 - x1)

m = (-3 - 3) / (2 - (-1))

m = -6 / 3 = -2

Substituting value of m in equation (1), we get

y - 3 = -2 ( x + 1)

y - 3 = -2x - 2

y = -2x + 1

2x + y = 1

Therefore, the equation of a line passing through the points (-1, 3) and (2, -3) is 2x + y = + 1.

2 Answers By Expert Tutors

Point-slope is y-y1=m(x-x1). If we plug the given values in, we have y-(-2)=-2/3(x-1). Simplifying gives y+2=-2/3x+2/3 or y=-2/3x-4/3. If we multiply everything by 3, we can get rid of the pesky denominators. Thus 3y=-2x-4. Add the x-term to get into standard form and you'll have 2x+3y=-4. There's your answer.

Chyke C. answered • 01/10/20

Patient and Knowledgeable Math & Science Tutor

For this type of question when we're trying to determine the equation of a line, we first need to find the slope of the line and insert it along with a (x,y) coordinate into the Point-Slope Formula.

The point slope formula is,  y-y₁=m(x-x₁). We are given a coordinate point and the slope so we plug in our given values. We can label our given coordinate point ( 1, -2 ) x1 which is 1 and y1 which is -2. Furthermore, we plug x1, y1, and m (our given slope) which is -2/3 into our Point-Slope Formula.

We then get:

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

Simplifying we get:

1. y + 2 = - 2/3x + 2/3

We then subtract the 2 from the left side over to the right to form a complete equation and have it in the form of y = mx + b (Slope-Intercept Form):

1. y= -2/3x - 4/3

You can then multiply the entire equation by 3 to get rid of the 3's in the denominator:

1. 3y = -2x - 4

Lastly we can move the 2x over to have it in standard form, which is ( Ax + By = C):

1. 3y + 2x = - 4 , where this is your final answer.
2. Please let me know if there are any steps i can clarify for you.