Understanding Lines: Equations, Slopes, and Parallelism

Posted on Aug 23, 2024 in Mathematics

The Line as a Geometric Locus

The line is a group of points that can be enlarged infinitely in both directions. A straight line is also defined as “a geometric locus of a point that moves on the plane in a way that, when located in any two positions, its slope m results as the same” (Vázquez, 2007).

Example: Graphing a Line

On the plane, graph the line that passes through the points (-3, 4) and (6, 3).

Steps

1. Procedure: Graph the given points on the Cartesian plane.
2. Procedure: Graph a line that joins both points in the plane.

Answer

This is the graph of the line:

Slope of a Line

The slope of a line is defined as the rate of change between ordinates and abscissas. It is the difference between ordinates divided by the difference of the abscissas, expressed by the following formula:

Example: Calculating Slope

Obtain the slope of a line that passes through points (-2, 5) and (4, 8) and graph it.

Steps

1. Procedure: Apply the formula to obtain the slope using the ordinates and abscissas of the given points.
2. Procedure: Simplify the resulting fraction.

Answer

The slope is 1/2, and its graph is:

The Relation Between Inclination Angle and the Slope of a Line

The Greek letter Δ (delta) represents change. The slope of a straight line is also known as a tangent, therefore:

To calculate the slope, consider the following:

1. When the slope m is positive, then 0° 0.
2. When the slope m is negative, then 90°
3. When the straight line is horizontal (Δy = 0), the line has a slope of 0, meaning it has no inclination.
4. When the straight line is vertical (Δx = 0), the line has no slope or is undefined.

Example: Finding the Inclination Angle

Find the angle of inclination between the x-axis and the line that passes through the points (2, 3) and (5, 4).

Steps

1. Procedure: First, obtain the slope of the line.
2. Procedure: To obtain the angle of inclination, substitute the slope in the following formula:

Answer

The inclination angle is 18.43°.

Equation of Lines Parallel to Coordinate Axes

The equations of lines parallel to the coordinate axes can be expressed as:

1. x = constant: This line is vertical and parallel to the y-axis; all its points are at the same distance from the y-axis.
2. y = constant: This line is horizontal and parallel to the x-axis; all its points are at the same distance from the x-axis.

Example: Graphing Lines Parallel to Axes

Graph the following lines in the Cartesian plane: x = 3, y = 5, x = -5, y = -1

Steps

1. Procedure: From the given equation, locate the value of x or y in the Cartesian plane.
2. Procedure: Graph a straight line that passes through the located point, so that the value of x or y in each line is always the same.

Types of Equations of the Line

The general equation of the line is expressed as: Ax + By + C = 0, where A, B, and C are real numbers.

Point-Slope Equation

Let A(x₁, y₁) be a known point that belongs to a line, m be the line’s slope, and P(x, y) be another point of the line. The following equation determines the slope of the AP line: m = (y – y₁) / (x – x₁). This expression is known as the point-slope equation. To obtain it, we need to know the point where the line passes and its slope or two points through which the line passes.

Example: Using Point-Slope Equation

Find the equation of the line that passes through point (4, 3) and has a slope of -2/3.

Steps

1. Procedure: Substitute the given information in the following equation: y – 3 = (-2/3)(x – 4)
2. Procedure: Number 3 passes multiplying to the other side of the equality. 3(y – 3) = -2(x – 4)
3. Procedure: To maintain x positive, we pass all terms from the left side to the right side and then we can obtain the general equation of the line. 3y – 9 = -2x + 8 => 2x + 3y – 17 = 0

Answer

Simplifying the previous equation, we obtain: 2x + 3y – 17 = 0

Equation of the Line Passing Through Two Points

Let A and B be two points of a line. From the definition of the slope of a line: m = (y₂ – y₁) / (x₂ – x₁). Substituting in the point-slope equation of the line, we obtain: (y – y₁) / (x – x₁) = (y₂ – y₁) / (x₂ – x₁). This equation is known as the equation of the line that passes through two points. You can substitute the data directly in this equation or calculate the slope first with both points and then substitute it in the point-slope equation by using any of the two known points.

Example: Using Two-Point Equation

Find the equation of the line that passes through points: A(3, -2) and B(5, 1).

Steps

1. Procedure: Calculate the slope of the line that passes through both points. m = (1 – (-2)) / (5 – 3) = 3/2
2. Procedure: With any of the two points from the line and the slope, substitute in the point-slope equation. In this case, we will select point A. y – (-2) = (3/2)(x – 3)
3. Procedure: Number 2 that is dividing on the right side passes multiplying on the left side. 2(y + 2) = 3(x – 3)
4. Procedure: To let x be positive, all terms on the left side are passed to the right side of the equation. Then we can obtain the general equation. 2y + 4 = 3x – 9 => 3x – 2y – 13 = 0

Answer

Simplifying the equation, the answer is: 3x – 2y – 13 = 0

Slope-Intercept Equation

Consider the line that passes through two points P(x, y) and Q(0, b), where point b is the one that crosses the y-axis. To calculate the slope between these two points: m = (y – b) / (x – 0) = (y – b) / x. When solving for y from this equation: y = mx + b, where b is the point where the line cuts the y-axis (its ordinate with the origin) and m is the line’s slope.

Example: Using Slope-Intercept Equation

Given the equation of the line 5x – 3y = 8, determine its slope and the point at which the line cuts the y-axis.

Steps

1. Procedure: To determine the slope and the intersection with the y-axis ordinate, you need to rearrange the equation into a slope-ordinate form. -3y = -5x + 8
2. Procedure: Solve for y in the given equation. y = (-5x + 8) / -3
3. Procedure: Divide each term of the numerator by the denominator. y = (5/3)x – 8/3

Answer

After applying the rules of signs, the final answer is: y = (5/3)x – 8/3. The slope is 5/3 and the ordinate to the origin is -8/3.

Symmetric Equation

Let the points be P(a, 0) and Q(0, b). To calculate the slope of the line: m = (b – 0) / (0 – a) = -b/a. Substitute this value into the equation in slope-ordinate form: y = (-b/a)x + b. Multiply everything by a: ay = -bx + ab. From dividing all terms by ab: x/a + y/b = 1. The above equation corresponds to the symmetric equation of the line.

Example: Using Symmetric Equation

Let a = -5 and b = 3 be the abscissa and the ordinate of a line, respectively. Determine the symmetric equation of the line with the given conditions.

Steps

1. Procedure: Substitute the given data by the symmetric equation of the line. x/-5 + y/3 = 1

Answer

The final equation is: x/-5 + y/3 = 1

Conditions of Parallelism and Perpendicularity According to Slope

Perpendicularity Theorem

Two lines are perpendicular if, and only if, their slopes are negative reciprocals of each other; that is, if m₁ is the slope of l₁ and m₂ is the slope of l₂, then it is true that: m₁ = -1/m₂

Example: Finding the Slope of a Perpendicular Line

Find the slope of the line that is perpendicular to the line 5x + 8y = 3

Steps

1. Procedure: Rearrange the equation in slope-ordinate form. 8y = -5x + 3 => y = (-5/8)x + 3/8
2. Procedure: The coefficient of x belongs to the slope of the line. m = -5/8
3. Procedure: Since the slope of the line should be perpendicular to this line, it should have a reciprocal slope and the opposite sign. So, the resulting slope is: m_{perpendicular} = 8/5

Answer

The slope of the line is 8/5.

Parallelism Theorem

Two lines are parallel if and only if their slopes are the same (m₁ = m₂) or collinear.