Find equation of straight line: points, slope, formula

This worksheet explains how to find the equation of a straight line. There are three common ways to find equation of straight line in the form y = mx + c.

By:

• finding the slope (m) and y-intercept (c),
• using the two-point method, or
• using the point-slope formula.

The straight line formula: y = mx + c

A straight line is given by the general equation y = mx + c, where:

• y represents values in the y axis
• m represents the slope (also called the slope)
• x represents the values on the x-axis
• c represents the y-intercept

How to find equation of straight line (given slope and a point on the line)

If you know the slope of a line and a point on the line, you can find the equation of the line by following the four steps.

1: Substitute the slope of the line for m in the equation y = mx + c.

2: Substitute the coordinates of a point on the line for the variables x and y in the equation y = mx + c.

3: Solve for c.

4: Substitute m and c into the equation y = mx + c to find the equation of a line.

Example: Find equation of the straight line with slope −5 and through the point (−2, −1).

• Step 1: Substitute the slope, which is −5, for m in the general equation y = mx + c.

y = −5 x + c

• Step 2: Substitute the coordinates of the point on the line for the variables x and y in the equation
  y = mx + c.

(−1) = −5(−2) + c

• Step 3: Solve for c.

(−1) = (−5)(−2) + c
(−1) = 10 + c
c = −11

• Step 4: Substitute m and c into the equation y = mx + c to find the equation of the line. The equation of the line is y = −5x – 11

Practice exercises: Find Equation of straight (given slope and a point on the line)

Find the equation of each of these lines.
1. Slope = 2 and point (4, −4)
2. Slope = 1 and point (1, 1)
3. Slope = ¹/₂ and point (0, −1)

How to find equation of straight line (through two points on the line)

If you know any two points, you can also find the equation of a line by following the 4 steps.

1: Find the slope of the line. Divide the change in y by the change in x.

2: Substitute the slope (m) into the equation y = mx + c.

3: Substitute one pair of coordinates into the equation for x and y and solve for c.

4: Substitute m and c into the equation y = mx + c to find the equation of a line.

Example: Find the equation of the line through the points (3, 4) and (6, 2).

• Step 1: Find the slope of the line. Subtract one set of coordinates from the other set of coordinates. The change in y is 4 − 2 or 2. The change in x is 3 − 6 or −3. Divide the change in y by the change in x to find the slope. The slope is – ²/₃.

• Step 2: Substitute the slope (m) into the general equation y = mx + c. y = − ²/₃ x + c
• Step 3: Substitute one pair of coordinates into the equation for x and y and solve for c.
  4 = – ²/₃ (3) + c
  4 = −2 + c
  c = 6
• Step 4: Substitute m and c into the equation y = mx + c. As a result, the equation of a line through the points (3, 4) and (6, 2) is y = – ²/₃ x + 6

Practice exercises: Finding Equation of straight line (through two points on the line)

Find the equation of the line through each pair of points.
1. (1, 1) and (6, 2)
2. (0, 5) and (2, 0)
3. (3, −1) and (4, −2)

Find equation of straight line (using the Point-slope Formula)

Another way to find the equation of a line is by using the point-slope formula.

Point-slope formula: y − y₁ = m(x − x₁) where x₁ and y₁ are the coordinates of a single point on the line, and m is the slope of the line.

Example 1: Find the equation of a line with slope 3 and point (2, 5).

Step 1: Substitute x = 2, y = 5, and m = 3 into the point-slope formula. y − 5 = 3(x − 5)
Step 2: Solve y − 5 = 3x − 15 y = 3x − 10

Example 2: Find the equation of a line with slope 1/2 and passing through the point (−4, 6).

Step 1 : Substitute x = −4, y = 6, and m = ¹/₂ into the point-slope formula y − 6 = ¹/₂ (x − (−4))
Step 2 : Solve
2.1. y − 6 = ¹/₂ (x + 4)
2.2. y − 6 = ¹/₂ x + 2
2.3. y = ¹/₂ x + 8

Practice exercise: Finding Equation of a straight line (using the Point-slope Formula)

Find the equation of a line with the given points and slopes.
1. point (3, 5) and slope −2?
2. point (0, 0) and slope 1?
3. point (−2, −6) and slope −1?
4. point (1, 1) and slope 0?

Answers to Questions on finding equations of straight lines

Finding Equation of a straight line given the slope and a point on the line
1. y = 2x – 12
2. y = x
3. y = ¹/₂ x – 1
Finding Equation of a straight line through two points
1. y = ¹/₅  x + ⁴/₅
2. y = − ⁵/₂ x + 5
3. y = −x + 2)
Finding Equation of a straight line (using the Point-Slope Formula)
1. y = −2x + 11
2. y = x
3. y = −x − 8
4. y = 1

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Key point: It is important to use the methods that teachers are introducing in class. Some maths classes may require students to use all three methods Equation of a Straight Line in this Worksheet.

Others may require a student to use one method only, so choose carefully which methods are applicable. Or see your mathematics teacher if you need help with finding out which method is recommended for you.

Downloadable Worksheet: Find Equation of Straight-lines Using Two Points, Slope and a Point and Point-slope Formula

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