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You have learned how to use the different representations of a line to calculate its \( x \)- and \( y \)-intercepts. These are the locations where a line crosses the horizontal and vertical axis, respectively.
Remember that you can represent a line using a graph or a table of values, or you can write an equation in slope-intercept form, \( y = mx + b \). But to write the equation of a line in slope-intercept form, you must have enough information to calculate both the line's slope and the location of its \( y \)-intercept.
Look at the table of values shown. The ordered pairs in this table represent a line. Do you have the information you need to write the equation of this line in slope-intercept form, using only the information given the table and not doing any additional work?
| \( x \) |
\( y \) |
|---|---|
\( - 15 \) |
\( 14 \) |
\( - 10 \) |
\( 11 \) |
\( - 5 \) |
\( 8 \) |
\( 5 \) |
\( 2 \) |
\( 10 \) |
\( - 1 \) |
No. While you can use two of the given ordered pairs to calculate the line's slope, you are not given the location of the \( y \)-intercept. Because you don't know the location of the line's \( y \)-intercept, you cannot use the slope-intercept form to write the equation without doing additional work.
If you do not know the value of a line's \( y \)-intercept, but you do know at least 2 ordered pairs that lie on the line, or you know the value the of the slope and one ordered pair that lies on the line, you can write the equation of the line using point-slope form.
The Point-Slope Equation of a Line
The point-slope form of a line is \( y - y_{1} = m(x - x_{1}) \).
In this equation:
- \( y \) is the dependent variable
- \( m \) is the slope
- \( x \) is the independent variable
- \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line
For Example
The equation \( y - 0 = 2(x - 1) \) is written in point-slope form.
Use the equation to identify the slope of this line and the coordinates of an ordered pair that lies on it.
The point-slope form of a line is \( y - y_{1} = m(x - x_{1}) \). |
In this form, \( m \) represents the slope, and \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line. |
Read the slope from the equation. |
\( y - y_{1} = \require{color}\colorbox{yellow}{$ m $}(x - x_{1}) \) \( y - 0 = \require{color}\colorbox{yellow}{$ 2 $}(x - 1) \) The slope of this line is \( m = 2 \). |
Read the coordinates of the ordered pair from the equation. |
\( y - \require{color}\colorbox{aqua}{$ y_{1} $} = m(x - \require{color}\colorbox{lime}{$ x_{1} $}) \) \( y - \require{color}\colorbox{aqua}{$ 0 $} = 2(x - \require{color}\colorbox{lime}{$ 1 $}) \) The coordinates of the ordered pair are \( (1,\ 0) \). Notice that since \( y_{1} = 0 \), this is the location of the line's \( x \)-intercept. |
Give it a shot!
How well can you read equations that are written in point-slope form? Use the activity below to practice your skills. Choose the best answer for each question.
The equation \( y - 6 = - 3(x - 4) \) is written in point-slope form. What is the value of this line's slope?
- \( m = - 3 \)
- \( m = 6 \)
- \( m = - 4 \)
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( m \) represents the slope of the line.
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( m \) represents the slope of the line.
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( m \) represents the slope of the line.
Which equation is written in point-slope form?
- \( ax^{2} + bx + c = 0 \)
- \( y = mx + b \)
- \( y - y_{1} = m(x - x_{1}) \)
The point slope form of an equation is \( y - y_{1} = m(x - x_{1}) \).
The point slope form of an equation is \( y - y_{1} = m(x - x_{1}) \).
The point slope form of an equation is \( y - y_{1} = m(x - x_{1}) \).
The equation \( y - 6 = - 3(x - 4) \) is written in point-slope form. Which represents the independent variable?
- \( x \)
- \( y \)
- \( x_{1} \)
The variable \( x \) is the independent variable.
The variable \( x \) is the independent variable.
The variable \( x \) is the independent variable.
The equation \( y - 6 = - 3(x - 4) \) is written in point-slope form. Name the coordinates of an ordered pair that lies on this line.
- \( (6, - 3) \)
- \( (4,\ 6) \)
- \( (- 4, - 6) \)
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line.
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line.
This equation is in the form \( y - y_{1} = m(x - x_{1}) \). In this form, \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line.
The equation \( y - 6 = - 3(x - 4) \) is written in point-slope form. Which represents the dependent variable?
- \( x \)
- \( y \)
- \( y_{1} \)
The variable \( y \) is the dependent variable.
The variable \( y \) is the dependent variable.
The variable \( y \) is the dependent variable.
Summary
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