EQUATION OF A LINE IN STANDARD FORM

Examples 1-4 : Write the given equations of lines in standard form :

Example 1 :

y = -3x + 2

Solution :

y = -3x + 2

Add 3x to both sides.

3x + y = 2

Subtract 2 from both sides.

3x + y - 2 = 0

Example 2 :

y = 2x - 5

Solution :

y = 2x - 5

Subtract 2x from both sides.

-2x + y = -5

Multiply both sides by -1.

2x - y = 5

Example 3 :

y = 3x/2 - 1

Solution :

y = 3x/2 - 1

Multiply both sides of the equation 2.

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

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

2y = 3x - 2

Subtract 3x from both sides.

-3x + 2y = -2

Multiply both sides by -1.

3x - 2y = 2

Example 4 :

y = 2x/3 - 1/2

Solution :

y = 2x/3 - 1/2

In the equation above, we find two different denominators 3 and 2.

The least common multiple of (3, 2) = 6.

Multiply both sides of the equation by 6 to get rid of the denominators 3 and 2.

6(y) = 6(2x/3 - 1/2)

6y = 6(2x/3) + 6(-1/2)

6y = 2(2x) + 3(-1)

6y = 4x - 3

Subtract 4x from both sides.

-4x + 6y = -3

Multiply both sides of the equation by -1.

4x - 6y = 3

Examples 5-6 : Write the equations of the given lines in standard form :

Example 5 :

Solution :

Formula to find the slope of a line when two points are given :

Substitute (x₁, y₁) = (-4, -4) and (x₂, y₂) = (5, 2).

Equation of the line in slope-intercept form :

y = mx + b

Substitute m = 2/3.

y = 2x/3 + b ----(1)

Substitute one of the two points on the line into the above equation to solve for b.

Substitute (5, 2) into the above equation.

2 = 2(5)/3 + b

2 = 10/3 + b

Subtract 10/3 from both sides.

2 - 10/3 = b

-4/3 = b

Substitute b = -4/3 in (1).

y = 2x/3 - 4/3

Multiply both sides by 3 to get rid of the denominators.

3(y) = 3(2x/3 - 4/3)

3y = 3(2x/3) + 3(-4/3)

3y = 2x - 4

Subtract 2x from both sides.

-2x + 3y = -4

Multiply both sides of the equation by -1.

2x - 3y = 4

Example 6 :

Solution :

Formula to find the slope of a line when two points are given :

Substitute (x₁, y₁) = (-4, 2) and (x₂, y₂) = (4, -4).

Equation of the line in slope-intercept form :

y = mx + b

Substitute m = -3/4.

y = -3x/4 + b ----(1)

Substitute one of the two points on the line into the above equation to solve for b.

Substitute (-4, 2) into the above equation.

2 = -3(-4)/4 + b

2 = -3(-1) + b

2 = 3 + b

Subtract 3 from both sides.

-1 = b

Substitute b = -1 in (1).

y = -3x/4 - 1

Multiply both sides by 4.

y = -3x/4 - 1

4(y) = 4(-3x/4 - 1)

4(y) = 4(-3x/4) + 4(-1)

4y = -3x - 4

Add 3x to both sides.

3x + 4y = -4

Example 7 :

Write the equation of a line in standard form that passes through the point (0, -1) and has slope -3.

Solution :

Equation of the line in slope-intercept form :

y = mx + b

Given : Slope is -3. So, substitute m = -3.

y = -3x + b ----(1)

Since the line passes through the point (0, -1) substitute the point (0, -1) into the above equation.

-1 = -3(0) + b

-1 = 0 + b

-1 = b

Substitute b = -1 in (1).

y = -3x + b

y = -3x - 1

Add 3x both sides.

3x + y = -1

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