Advanced Algebra Notes
Section 2.4: Write Equations of Lines
Given slope and y-intercept b
Use slope-intercept form
𝑦 = 𝑚𝑥 + 𝑏
Given slope and a point (𝑥, 𝑦)
Use point slope form
𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1
Given points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 )
First use the slope formula to find m.
𝑅𝑖𝑠𝑒 𝑜𝑣𝑒𝑟 𝑅𝑢𝑛 𝑔𝑒𝑡𝑠 𝑡ℎ𝑒 𝑗𝑜𝑏 𝑑𝑜𝑛𝑒! Then use point-slope form with either point
𝑅𝑖𝑠𝑒 (𝑦)
𝑈𝑠𝑒 𝑠𝑙𝑜𝑝𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑚 (𝑠𝑙𝑜𝑝𝑒)
𝑚=
𝑇ℎ𝑒𝑛 𝑒𝑖𝑡ℎ𝑒𝑟 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑤𝑖𝑙𝑙 𝑤𝑜𝑟𝑘!
𝑅𝑢𝑛 (𝑥)
Example 1
Write an equation of the line shown.
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑙𝑖𝑛𝑒
𝑦 = 𝑚𝑥 + 𝑏
𝑊ℎ𝑎𝑡 𝑖𝑠 𝑜𝑢𝑟 𝑠𝑙𝑜𝑝𝑒? 𝑊ℎ𝑒𝑟𝑒
𝑑𝑜𝑒𝑠 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑠𝑡𝑎𝑟𝑡 𝑜𝑛 𝑡ℎ𝑒
𝑦 − 𝑎𝑥𝑖𝑠?
3
𝑦 = 𝑥 + (−2)
4
Example 2
Write an equation of the line that passes through (5,4) and has a slope of −3.
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
𝑃𝑜𝑖𝑛𝑡 − 𝑠𝑙𝑜𝑝𝑒 𝑓𝑜𝑟𝑚
𝑃𝑙𝑢𝑔 𝑖𝑛 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒𝑠!
𝑦 − 4 = −3(𝑥 − 5)
𝑦 − 4 = −3(𝑥 − 5)
𝑦 − 4 = −3𝑥 + 15
𝑦 = −3𝑥 + 19
Example 3
Write an equation of the line that passes through (−2,3) and is
Parallel to the line 𝑦 = −4𝑥 + 1
𝐿𝑖𝑛𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑠𝑙𝑜𝑝𝑒!
𝑦 = −4𝑥 + 𝑏
−5 = 𝑏
𝑦 = −4𝑥 − 5
3 = −4(−2) + 𝑏
Perpendicular to the line 𝑦 = −4𝑥 + 1
𝐿𝑖𝑛𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑎𝑣𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑟𝑒𝑐𝑖𝑝𝑟𝑜𝑐𝑎𝑙 𝑠𝑙𝑜𝑝𝑒!
1
1
𝑦 − 𝑦1 = −4(𝑥 − 𝑥1 )
1
7
𝑦 = 𝑥+ +3
𝑦= 𝑥+
1
4
2
4
2
𝑦 − 3 = (𝑥 + 2)
4
Example 4
Write an equation of the line that passes through (5, −2) and (2,10)
𝐹𝑖𝑟𝑠𝑡 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒!
10 − (−2)
𝑚=
2−5
12
𝑚=
−3
𝑚 = −4
𝑅𝑖𝑠𝑒 𝑜𝑣𝑒𝑟 𝑅𝑢𝑛 𝑔𝑒𝑡𝑠 𝑡ℎ𝑒 𝑗𝑜𝑏 𝑑𝑜𝑛𝑒!
𝑦 − 𝑦1 = −4(𝑥 − 𝑥1 )
𝐸𝑖𝑡ℎ𝑒𝑟 𝑝𝑜𝑖𝑛𝑡 𝑤𝑖𝑙𝑙 𝑤𝑜𝑟𝑘!
𝑦 − (−2) = −4(𝑥 − 5)
𝑦 + 2 = −4𝑥 + 20
𝑦 = −4𝑥 + 18
Example 5
In the school year ending in 1993, 2.00 million females participated in U.S.
high school sports. By 2003, the number had increased to 2.86 million. Write
a linear equation that models female sports participation.
𝑥 = # 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠
𝑦 = # 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑛𝑡𝑠
𝑦 = 𝑚𝑥 + 𝑏
𝑦 = (.086)𝑥 + 𝑏
𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑡𝑖𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑓𝑟𝑜𝑚 1993 𝑡𝑜 2003
… 𝑖𝑛 𝑜𝑡ℎ𝑒𝑟 𝑤𝑜𝑟𝑑𝑠 … 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒!
2.86 − 2.00
𝑚=
10 − 0
𝑚 = .086
2.00 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡
𝑦 = (.086)𝑥 + 2.00
Closing: How do you write an equation of a line?
𝑌𝑜𝑢 𝑢𝑠𝑒 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒, 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑎𝑛𝑑 𝑎 𝑔𝑖𝑣𝑒𝑛 𝑝𝑜𝑖𝑛𝑡
𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒
𝐷𝑒𝑐𝑖𝑑𝑒 𝑤ℎ𝑖𝑐ℎ 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑤𝑜𝑟𝑘𝑠 𝑏𝑒𝑠𝑡 …
𝑆𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝑦 = 𝑚𝑥 + 𝑏
𝑃𝑜𝑖𝑛𝑡 − 𝑠𝑙𝑜𝑝𝑒 𝑓𝑜𝑟𝑚
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
HOMEWORK
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