Equations Of Lines Birmingham AL

Equations involving one or two variables can be graphed on any x− y coordinate plane. In general, the following principles are true:

• If a point lies on the graph of an equation, then its coordinates make the equation a true statement.

• If the coordinates of a point make an equation a true statement, then the point lies on the graph of the equation.

A linear equation is any equation whose graph is a line. All linear equations can be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both zero. The following examples are linear equations and their respective A, B, and C values.

This form for equations of lines is known as the standard form for the equation of a line.

The x -intercept of a graph is the point where the graph intersects the x-axis. It always has a y-coordinate of zero. A horizontal line that is not the x-axis has no x-intercept .

The y -intercept of a graph is the point where the graph intersects the y-axis. It always has an x-coordinate of zero. A vertical line that is not the y-axis has no y-intercept .

One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of two points is necessary to graph a linear equation.

Example 1: Draw the graph of 2 x + 3 y = 12 by finding the x-intercept and the y-intercept .

The x-intercept has a y-coordinate of zero. Substituting zero for y, the resulting equation is 2 x+ 3(0) = 12. Now solving for x,

The x-intercept is at (6, 0), or the x-intercept value is 6.

The y-intercept has an x-coordinate of zero. Substituting zero for x, the resulting equation is 2(0) + 3 y = 12. Now solving for y,

The y-intercept is at (0, 4), or the y-intercept value is 4.

The line can now be graphed by graphing these two points and then drawing the line they determine (Figure 1 ).

Figure 1 Drawing the graph of a linear equation after finding the x-intercept and the y-intercept .

Example 2: Draw the graph of x = 2.

x = 2 is a vertical line whose x-coordinate is always 2 (Figure 2 )

Figure 2 The graph of a vertical line.

Example 3: Draw the graph of y = −1.

y = −1 is a horizontal line whose y-coordinate is always −1. See Figure 3 .

Figure 3

Suppose that A is a particular point called ( x₁, y₁) and B is any point called ( x, y). Then the slope of the line through A and B is represented by

Applying the Cross-Products Property, y − y₁ = m ( x − x₁). This is the point-slope form of a nonvertical line.

Theorem107: The point-slope form of a line passing through ( x₁, y₁) and having slope m is y − y₁ = m ( x − x₁).

Example 4: Find the equation of a line containing the points (−3,4) and (7,2) and write the equation in (a) point-slope form and (b) standard form.

1. (a) For the point-slope form, first find the slope, m.
   

   
   

   Now choose either original point—say, (−3, 4).

   So,
   

   
   

2. (b) Begin with the point-slope form and clear it of fractions by multiplying both sides by the least common denominator.
   

   
   

   Multiply both sides by 5.
   

   
   

Get x and y on one side and the constants on the other side by adding x to both sides and adding 20 to both sides.

A nonvertical line written in standard form is Ax + By = C with B ≠ 0. If this equation is solved for y, it becomes

Let b denote the y-intercept of a line. The point-slope form of the equation of the line passing through (0, b) with slope m is

Adding b to both sides of the equation yields

This is known as the slope-intercept form of the equation of a nonvertical line. Note that, in order to obtain the slope-intercept form, a nonvertical line written in standard form Ax + By = C with B ≠ 0 can be solved algebraically for y.

Theorem 108: The slope-intercept form of a nonvertical line with slope m and y-intercept value b is y = mx + b.

Example 5: Find the slope and y-intercept value of the line with equation 3 x − 4 y = 20.

Therefore, the slope of the line is 3/4 and the y-intercept value is −5.

Example 6: Line l₁ has equation 2 x + 5 y = 10. Line l₂ has equation 4 x + 10 y = 30. Line l₃ has equation 15 x − 6 y = 12. Which lines, if any, are parallel?

Put each equation into slope-intercept form and determine the slope of each line.

• l₁:
  

  
  

• l₂:
  

  
  

• l₃:
  

  
  

Slope l₁ = slope l₂, therefore; l₁ // l₂ by Theorem 104.

Because (slope l₁)(slope l₃) = −1 and (slope l₂)(slope l₃) = −1, l₁ ⊥ l₃ and l₂ ⊥ l₃ by Theorem 106.

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