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Introduction

Definition

The equation of a straight line is a mathematical representation of the relationship between two variables, x and y, that can be graphically represented as a straight line. The equation of a straight line is usually written in the form y = mx + c, where m is the slope of the line, and c is the y-intercept. The slope of a line is a measure of how steeply it rises or falls, and the y-intercept is the point where the line crosses the y-axis.

Deriving the Equation of a Straight Line

The equation of a straight line can be derived using two methods: the point-slope form and the slope-intercept form.

Point-Slope Form

The point-slope form of the equation of a straight line is given by y – y1 = m(x – x1), where m is the slope of the line and (x1, y1) is a point on the line. To derive this formula, we use the fact that the slope of a line is given by the change in y divided by the change in x, or m = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are two points on the line. Rearranging this formula gives us y – y1 = m(x – x1).

Slope-Intercept Form

The slope-intercept form of the equation of a straight line is given by y = mx + c, where m is the slope of the line, and c is the y-intercept. To derive this formula, we start with the point-slope form and simplify it by solving for y. We get y – y1 = m(x – x1), which can be rearranged to give y = mx – mx1 + y1. We can then substitute the y-intercept for y1 and simplify to get y = mx + c, where c = y1 – mx1.

Forms of the Equation of a Straight Line

There are three different forms of the equation of a straight line: the standard form, the slope-intercept form, and the point-slope form.

Standard Form

The standard form of the equation of a straight line is Ax + By = C, where A, B, and C are constants. This form is useful for finding the x- and y-intercepts of a line and for solving systems of linear equations. To convert the slope-intercept form or the point-slope form to the standard form, we simply rearrange the terms.

Slope-Intercept Form

The slope-intercept form of the equation of a straight line is y = mx + c, where m is the slope of the line, and c is the y-intercept. This form is useful for graphing a line and for finding the equation of a line when we know its slope and y-intercept.

Point-Slope Form

The point-slope form of the equation of a straight line is y – y1 = m(x – x1), where m is the slope of the line, and (x1, y1) is a point on the line. This form is useful for finding the equation of a line when we know its slope and one point on.

Examples

Example 1: Find the equation of a line that passes through the points (2, 3) and (4, 7).

Using the point-slope form, we have:

m = (7 – 3)/(4 – 2) = 2 y – 3 = 2(x – 2) y = 2x – 1

Therefore, the equation of the line is y = 2x – 1.

Example 2: Find the equation of a line that passes through the point (-1, 5) and has a slope of -3.

Using the point-slope form, we have:

m = -3 y – 5 = -3(x + 1) y = -3x + 2

Therefore, the equation of the line is y = -3x + 2.

Example 3: Find the equation of a line that passes through the point (5, -2) and is parallel to the line y = 2x + 3.

Since the line we want to find is parallel to y = 2x + 3, its slope is also 2. Using the point-slope form, we have:

m = 2 y – (-2) = 2(x – 5) y = 2x – 12

Therefore, the equation of the line is y = 2x – 12.

Example 4: Find the equation of a line that passes through the point (3, 4) and is perpendicular to the line y = -1/2x + 7.

Since the line we want to find is perpendicular to y = -1/2x + 7, its slope is the negative reciprocal of -1/2, which is 2. Using the point-slope form, we have:

m = 2 y – 4 = 2(x – 3) y = 2x – 2

Therefore, the equation of the line is y = 2x – 2.

Example 5: Find the equation of a line that passes through the points (-3, 4) and (1, 6).

Using the point-slope form, we have:

m = (6 – 4)/(1 – (-3)) = 1/2 y – 4 = 1/2(x – (-3)) y = 1/2x + 5

Therefore, the equation of the line is y = 1/2x + 5.

Example 6: Find the equation of a line that passes through the point (0, -2) and is parallel to the x-axis.

Since the line we want to find is parallel to the x-axis, its slope is 0. Using the point-slope form, we have:

m = 0 y – (-2) = 0(x – 0) y = -2

Therefore, the equation of the line is y = -2.

Example 7: Find the equation of a line that passes through the point (-5, 2) and is perpendicular to the y-axis.

Since the line we want to find is perpendicular to the y-axis, its slope is undefined. This means that the equation of the line is x = -5.

Example 8: Find the equation of a line that passes through the point (-2, 3) and has a slope of 0.

Since the slope is 0, the line is horizontal. Using the point-slope form, we have:

m = 0 y – 3 = 0(x – (-2)) y = 3

Therefore, the equation of the line is y = 3.

Example

9: Find the equation of a line that passes through the point (-4, -1) and is perpendicular to the line y = 3x – 2.

Since the line we want to find is perpendicular to y = 3x – 2, its slope is the negative reciprocal of 3, which is -1/3. Using the point-slope form, we have:

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

Therefore, the equation of the line is y = -1/3x + 1.

Example 10: Find the equation of a line that passes through the points (1, 2) and (1, 6).

Since the line is vertical and passes through the point (1, 2), its equation is x = 1.

FAQ

1. What is the point-slope form of the equation of a line? The point-slope form of the equation of a line is y – y1 = m(x – x1), where (x1, y1) is a point on the line and m is the slope of the line.
2. What is the slope-intercept form of the equation of a line? The slope-intercept form of the equation of a line is y = mx + b, where m is the slope of the line and b is the y-intercept of the line.
3. How do you find the slope of a line? The slope of a line can be found by dividing the change in y by the change in x between two points on the line. This can be represented by the formula m = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are two points on the line.
4. What does it mean for two lines to be parallel? Two lines are parallel if they have the same slope. This means that they never intersect and are always the same distance apart.
5. What does it mean for two lines to be perpendicular? Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that they intersect at a 90-degree angle.

Quiz

1. What is the equation of a line that passes through the points (2, 3) and (4, 7)? a) y = 2x – 1 b) y = 3x + 1 c) y = 2x + 3 d) y = -2x + 7
2. What is the equation of a line that passes through the point (-1, 5) and has a slope of -3? a) y = 3x + 2 b) y = -3x + 8 c) y = -3x + 2 d) y = 3x + 8
3. What is the equation of a line that passes through the point (5, -2) and is parallel to the line y = 2x + 3? a) y = 2x + 12 b) y = -2x – 12 c) y = -2x + 12 d) y = 2x – 12
4. What is the equation of a line that passes through the point (3, 4) and is perpendicular to the line y = -1/2x + 7? a) y = -1/2x – 1/2 b) y = 2x – 2 c) y = -2x + 2 d) y = 1
5. What is the slope of a line that passes through the points (2, 5) and (6, 9)? a) 2 b) 4 c) 1/2 d) 1/4
6. What is the equation of a line that passes through the points (-3, 1) and (3, -1)? a) y = -x + 2 b) y = x c) y = -x – 2 d) y = x – 2
7. What is the equation of a line that passes through the point (0, 4) and is perpendicular to the line y = 1/2x – 3? a) y = -2x + 4 b) y = 2x – 4 c) y = -2x – 4 d) y = 2x + 4
8. What is the equation of a line that passes through the point (2, -3) and has a slope of 5? a) y = 5x – 13 b) y = -5x – 13 c) y = 5x + 13 d) y = -5x + 13
9. What is the equation of a line that passes through the point (-4, -1) and is perpendicular to the line y = 3x – 2? a) y = -1/3x + 1 b) y = 3x – 13 c) y = -3x + 1 d) y = 1/3x + 1
10. What is the equation of a line that passes through the points (1, 2) and (1, 6)? a) y = 4x – 2 b) y = -4x – 2 c) x = 1 d) y = 2x + 1

Answers:

1. d) y = -2x + 7
2. c) y = -3x + 2
3. d) y = 2x – 12
4. b) y = 2x – 2
5. b) 4
6. c) y = -x – 2
7. a) y = -2x + 4
8. c) y = 5x + 13
9. a) y = -1/3x + 1
10. c) x = 1