Equation of a Line - Revision history

Khanh: /* Two-point form */

2022-01-11T21:27:08Z

Two-point form

Khanh at 21:23, 11 January 2022

2022-01-11T21:23:43Z

Khanh at 21:21, 11 January 2022

2022-01-11T21:21:21Z

Khanh at 21:04, 11 January 2022

2022-01-11T21:04:41Z

Khanh: Created page with "* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb..."

2021-09-22T23:10:52Z

Created page with "* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy * [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb..."

New page

* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy
* [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:writing-slope-intercept-equations/v/equation-of-a-line-1 Slope-intercept equation from slope & point], Khan Academy

@@ Line 37: / Line 37: @@
 ===Two-point form===
-Given two different points {{math|(''x''{{sub|1}}, ''y''{{sub|1}})}} and {{math|(''x''{{sub|2}}, ''y''{{sub|2}})}}, there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
+Given two different points (''x''<sub>1</sub>, ''y''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>), there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
-If {{math|''x''{{sub|1}} ≠ ''x''{{sub|2}}}}, the slope of the line is <math>\frac{y_2 - y_1}{x_2 - x_1}.</math> Thus, a point-slope form is<ref name=WilsonTracey />
+If ''x''<sub>1</sub> ≠ ''x''<sub>2</sub>, the slope of the line is <math>\frac{y_2 - y_1}{x_2 - x_1}.</math> Thus, a point-slope form is
 :<math>y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).</math>
 By clearing denominators, one gets the equation
 :<math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,</math>
-which is valid also when {{math|1=''x''{{sub|1}} = ''x''{{sub|2}}}} (for verifying this, it suffices to verify that the two given points satisfy the equation).
+which is valid also when ''x''<sub>1</sub> = ''x''<sub>2</sub> (for verifying this, it suffices to verify that the two given points satisfy the equation).
 This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:

@@ Line 2: / Line 2: @@
 There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
-====Slope–intercept form or Gradient-intercept form====
+===Slope–intercept form or Gradient-intercept form===
 A non-vertical line can be defined by its slope {{mvar|m}}, and its {{mvar|y}}-intercept ''y''<sub>0</sub> (the {{mvar|y}} coordinate of its intersection with the {{mvar|y}}-axis). In this case its ''linear equation'' can be written
 :<math>y=mx+y_0.</math>

@@ Line 1: / Line 1: @@
-===Equation of a line===
+== Equation of a line ==
 There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
 ====Slope–intercept form or Gradient-intercept form====
-A non-vertical line can be defined by its slope {{mvar|m}}, and its {{mvar|y}}-intercept {{math|''y''{{sub|0}}}} (the {{mvar|y}} coordinate of its intersection with the {{mvar|y}}-axis). In this case its ''linear equation'' can be written
+A non-vertical line can be defined by its slope {{mvar|m}}, and its {{mvar|y}}-intercept ''y''<sub>0</sub> (the {{mvar|y}} coordinate of its intersection with the {{mvar|y}}-axis). In this case its ''linear equation'' can be written
 :<math>y=mx+y_0.</math>
-If, moreover, the line is not horizontal, it can be defined by its slope and its {{mvar|x}}-intercept {{math|''x''{{sub|0}}}}. In this case, its equation can be written
+If, moreover, the line is not horizontal, it can be defined by its slope and its {{mvar|x}}-intercept ''x''<sub>0</sub>. In this case, its equation can be written
 :<math>y=m(x-x_0),</math>
 or, equivalently,
 :<math>y=mx-mx_0.</math>
-These forms rely on the habit of considering a non vertical line as the [[graph of a function]].<ref>{{harvnb|Larson|Hostetler|2007|loc=p. 25}}</ref> For a line given by an equation
+These forms rely on the habit of considering a non vertical line as the graph of a function. For a line given by an equation
 :<math>ax+by+c = 0,</math>
 these forms can be easily deduced from the relations
@@ Line 20: / Line 20: @@
 \end{align}</math>
-====Point–slope form or Point-gradient form====
+===Point–slope form or Point-gradient form===
 A non-vertical line can be defined by its slope {{mvar|m}}, and the coordinates <math>x_1, y_1</math> of any point of the line. In this case, a linear equation of the line is
@@ Line 31: / Line 31: @@
 for emphasizing that the slope of a line can be computed from the coordinates of any two points.
-====Intercept form====
+===Intercept form===
-A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values {{math|''x''{{sub|0}}}} and {{math|''y''{{sub|0}}}} of these two points are nonzero, and an equation of the line is<ref name=WilsonTracey>{{harvnb|Wilson|Tracey|1925|loc=pp. 52-53}}</ref>
+A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values ''x''<sub>0</sub> and ''y''<sub>0</sub> of these two points are nonzero, and an equation of the line is
 :<math>\frac{x}{x_0} + \frac{y}{y_0} = 1.</math>
-(It is easy to verify that the line defined by this equation has {{math|''x''{{sub|0}}}} and {{math|''y''{{sub|0}}}} as intercept values).
+(It is easy to verify that the line defined by this equation has ''x''<sub>0</sub> and ''y''<sub>0</sub> as intercept values).
-====Two-point form====
+===Two-point form===
 Given two different points {{math|(''x''{{sub|1}}, ''y''{{sub|1}})}} and {{math|(''x''{{sub|2}}, ''y''{{sub|2}})}}, there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
@@ Line 42: / Line 42: @@
 :<math>y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).</math>
-By [[clearing denominators]], one gets the equation
+By clearing denominators, one gets the equation
 :<math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,</math>
 which is valid also when {{math|1=''x''{{sub|1}} = ''x''{{sub|2}}}} (for verifying this, it suffices to verify that the two given points satisfy the equation).
@@ Line 50: / Line 50: @@
 (exchanging the two points changes the sign of the left-hand side of the equation).
-====Determinant form====
+===Determinant form===
-The two-point form of the equation of a line can be expressed simply in terms of a [[determinant]]. There are two common ways for that.
+The two-point form of the equation of a line can be expressed simply in terms of a determinant. There are two common ways for that.
 The equation <math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0</math> is the result of expanding the determinant in the equation
@@ Line 63: / Line 63: @@
 \end{vmatrix}=0.</math>
-Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a [[hyperplane]] passing through {{mvar|n}} points in a space of dimension {{math|''n'' – 1}}. These equations rely on the condition of [[linear dependence]] of points in a [[projective space]].
+Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through {{mvar|n}} points in a space of dimension {{math|''n'' – 1}}. These equations rely on the condition of linear dependence of points in a projective space.
 == Licensing ==

← Older revision		Revision as of 21:04, 11 January 2022
Line 1:		Line 1:
		+	===Equation of a line===
		+	There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
		+
		+	====Slope–intercept form or Gradient-intercept form====
		+	A non-vertical line can be defined by its slope {{mvar\|m}}, and its {{mvar\|y}}-intercept {{math\|''y''{{sub\|0}}}} (the {{mvar\|y}} coordinate of its intersection with the {{mvar\|y}}-axis). In this case its ''linear equation'' can be written
		+	:<math>y=mx+y_0.</math>
		+
		+	If, moreover, the line is not horizontal, it can be defined by its slope and its {{mvar\|x}}-intercept {{math\|''x''{{sub\|0}}}}. In this case, its equation can be written
		+	:<math>y=m(x-x_0),</math>
		+	or, equivalently,
		+	:<math>y=mx-mx_0.</math>
		+
		+	These forms rely on the habit of considering a non vertical line as the [[graph of a function]].<ref>{{harvnb\|Larson\|Hostetler\|2007\|loc=p. 25}}</ref> For a line given by an equation
		+	:<math>ax+by+c = 0,</math>
		+	these forms can be easily deduced from the relations
		+	:<math>\begin{align}
		+	m&=-\frac ab,\\
		+	x_0&=-\frac ca,\\
		+	y_0&=-\frac cb.
		+	\end{align}</math>
		+
		+	====Point–slope form or Point-gradient form====
		+
		+	A non-vertical line can be defined by its slope {{mvar\|m}}, and the coordinates <math>x_1, y_1</math> of any point of the line. In this case, a linear equation of the line is
		+	:<math>y=y_1 + m(x-x_1),</math>
		+	or
		+	:<math>y=mx +y_1-mx_1.</math>
		+
		+	This equation can also be written
		+	:<math>y-y_1=m(x-x_1)</math>
		+	for emphasizing that the slope of a line can be computed from the coordinates of any two points.
		+
		+	====Intercept form====
		+	A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values {{math\|''x''{{sub\|0}}}} and {{math\|''y''{{sub\|0}}}} of these two points are nonzero, and an equation of the line is<ref name=WilsonTracey>{{harvnb\|Wilson\|Tracey\|1925\|loc=pp. 52-53}}</ref>
		+	:<math>\frac{x}{x_0} + \frac{y}{y_0} = 1.</math>
		+	(It is easy to verify that the line defined by this equation has {{math\|''x''{{sub\|0}}}} and {{math\|''y''{{sub\|0}}}} as intercept values).
		+
		+	====Two-point form====
		+	Given two different points {{math\|(''x''{{sub\|1}}, ''y''{{sub\|1}})}} and {{math\|(''x''{{sub\|2}}, ''y''{{sub\|2}})}}, there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
		+
		+	If {{math\|''x''{{sub\|1}} ≠ ''x''{{sub\|2}}}}, the slope of the line is <math>\frac{y_2 - y_1}{x_2 - x_1}.</math> Thus, a point-slope form is<ref name=WilsonTracey />
		+	:<math>y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).</math>
		+
		+	By [[clearing denominators]], one gets the equation
		+	:<math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,</math>
		+	which is valid also when {{math\|1=''x''{{sub\|1}} = ''x''{{sub\|2}}}} (for verifying this, it suffices to verify that the two given points satisfy the equation).
		+
		+	This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
		+	:<math>(y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1) =0</math>
		+	(exchanging the two points changes the sign of the left-hand side of the equation).
		+
		+	====Determinant form====
		+	The two-point form of the equation of a line can be expressed simply in terms of a [[determinant]]. There are two common ways for that.
		+
		+	The equation <math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0</math> is the result of expanding the determinant in the equation
		+	:<math>\begin{vmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{vmatrix}=0.</math>
		+
		+	The equation <math> (y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1)=0</math> can be obtained be expanding with respect to its first row the determinant in the equation
		+	:<math>\begin{vmatrix}
		+	x&y&1\\
		+	x_1&y_1&1\\
		+	x_2&y_2&1
		+	\end{vmatrix}=0.</math>
		+
		+	Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a [[hyperplane]] passing through {{mvar\|n}} points in a space of dimension {{math\|''n'' – 1}}. These equations rely on the condition of [[linear dependence]] of points in a [[projective space]].
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Linear_equation Linear equation, Wikipedia] under a CC BY-SA license
		+
		+	== Resources ==
	* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy		* [https://www.youtube.com/watch?v=gvwKv6F69F0 Equation of a Line], Khan Academy
	* [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:writing-slope-intercept-equations/v/equation-of-a-line-1 Slope-intercept equation from slope & point], Khan Academy		* [https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:writing-slope-intercept-equations/v/equation-of-a-line-1 Slope-intercept equation from slope & point], Khan Academy