{"id":9404,"date":"2023-05-03T10:00:42","date_gmt":"2023-05-03T04:30:42","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=9404"},"modified":"2023-05-04T09:49:10","modified_gmt":"2023-05-04T04:19:10","slug":"equations-straight-lines","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/equations-straight-lines\/","title":{"rendered":"Equations of Straight Lines"},"content":{"rendered":"

Equations of Straight Lines<\/strong><\/span><\/h2>\n

Depending upon the given information, equations of lines can take on several forms:<\/p>\n

Slope Intercept Form:<\/strong>
\ny = mx + b
\nUse this form when you know, or can find, the slope, m, and the y-intercept, b.<\/p>\n

Point Slope Form:<\/strong>
\ny – y1<\/sub> = m(x – x1<\/sub>)
\nUse this form when you know, or can find, a point on the line (x1<\/sub>, y1<\/sub>), and the slope, m.<\/p>\n

Standard Form:<\/strong>
\nAx + By = C
\nThe A and B values in this form cannot be zero. Use when asked to state the answer in Standard Form.
\nMay also be Ax + By – C = 0.<\/p>\n

Horizontal Line Form:<\/strong>
\ny = 7 (or any Real number)
\nLines that are horizontal have a slope of zero. They have “run”, but no “rise”. The rise\/run formula for slope always yields zero since rise = 0. Every point on this line has a y-value of 7. When writing the equation, we have
\ny = mx + b
\ny = 0x + 7
\ny = 7.
\nNote: The equation of the x-axis is y = 0.<\/p>\n

Vertical Line Form:<\/strong>
\nx = -5 (or any Real number)
\nLines that are vertical have no slope (it does not exist, undefined). They have “rise”, but no “run”. The rise\/run formula for slope always has a zero denominator and is undefined. Every point on this line has an x-value of -5.
\nNote: The equation of the y-axis is x = 0.
\nNote:
\nLines that are parallel have equal slopes.
\nLines that are perpendicular have negative reciprocal slopes.
\n(A line with m = 4 will be perpendicular to a line with m = -\u00bc)<\/p>\n

Examples for Equations of Lines<\/strong>
\nHere are a few of the more common types of problems involving the equations of lines.<\/p>\n

Example 1:<\/strong> Find the slope and y-intercept of the equation 2y = 8x – 11.
\nSolution: First, solve for “y =”.
\ny = 4x – 5.5
\ny = mx + b
\nThe slope, m, is 4.
\nThe y-intercept, b, is -5.5.<\/p>\n

Example 2:<\/strong> Find the equation of a line whose slope is -2 and who crosses the y-axis at (0,-3).
\nSolution:<\/strong> The m = -2 and b = -3.
\ny = mx + b
\ny = -2x + (-3)
\ny = -2x – 3<\/p>\n

Example 3:<\/strong> Find the equation of a line whose slope is 4 and passes through the point (-3,5).
\nSolution:<\/strong> The m = 4 and (x1<\/sub>, y1<\/sub>) = (-3,5).
\nPoint-Slope Form
\ny – y1<\/sub> = m(x – x1<\/sub>)
\ny – 5 = 4(x – (-3))
\ny – 5 = 4(x + 3)
\ny – 5 = 4x + 12
\ny = 4x + 17<\/p>\n

Example 4:<\/strong> Find the equation of a line passing through the points (-2,6) and (-4,-2).
\nSolution:<\/strong> Find slope first.
\nm = (6 – (-2))\/(-2 – (-4)) = 8\/2 = 4
\nUse either point as (x1<\/sub>, y1<\/sub>): (-2,6).
\nPoint-Slope Form
\ny – y1<\/sub> = m(x – x1<\/sub>)
\ny – 6 = 4(x – (-2))
\ny – 6 = 4(x + 2)
\ny – 6 = 4x + 8
\ny = 4x + 14<\/p>\n

Example 5:<\/strong> Find the equation of a line that is parallel to the line y = -2x + 8 and passes through the point (3,6).
\nSolution:<\/strong> Parallel means equal slopes.
\nSo, m = -2 and (x1<\/sub>, y1<\/sub>) = (3,6).
\nPoint-Slope Form
\ny – y1<\/sub> = m(x – x1<\/sub>)
\ny – 6 = -2(x – 3)
\ny – 6 = -2x + 6)
\ny = -2x + 12<\/p>\n

Example 6:<\/strong> Find the equation of a line that is perpendicular to the line y = x + 7 and has the same y-intercept as 3y = 2x – 9. State the answer in Standard Form.
\nSolution:<\/strong> Perpendicular means negative reciprocal slopes.
\nSo, m = -1 and b = -3.
\ny = mx + b
\ny = -1x + (-3)
\ny = -x – 3
\nx + y = -3 (Standard Form)<\/p>\n