Section 3.3. Equation of a line

Section 3.3: Equation of a line Chapter 3: Lines The slope–intercept form of an equation of a line with slope m and y-intercept, (0, 𝑏) is: 𝑦 = 𝑚𝑥 + 𝑏 How to find an equation of a line given the 𝑠𝑙𝑜𝑝𝑒 𝑎𝑛𝑑 𝑦 − 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 using the 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form. Step 1. Step 2. Step 3. Step 4. Identify the slope. Identify the point. Substitute the values into slope-intercept form, 𝑦 = 𝑚𝑥 + 𝑏 and solve for b. Write your answer by replacing the values of m and b in the y = mx+ b equation. For Example: Use the slope intercept formula to find the 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form of an equation of a line with 𝑠𝑙𝑜𝑝𝑒 𝑚 = 6 𝑎𝑛𝑑 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (−2,5). Write your answer in 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form. Step 1. 𝑆𝑙𝑜𝑝𝑒: 𝑚 = 6 Step 2. 𝑃𝑜𝑖𝑛𝑡 (−2, 5) Step 3. Let 𝑥 = −2, 𝑦 = 5 𝑎𝑛𝑑 𝑚 = 6 in the equation 𝑦 = 𝑚𝑥 + 𝑏 and solve for 𝑏. 5 = 6(−2) + 𝑏 5 = −12 + 𝑏 +12 + 12 17 = 𝑏 Step 4. Replace 𝑚 𝑤𝑖𝑡ℎ 6 𝑎𝑛𝑑 𝑏 𝑤𝑖𝑡ℎ 17 in the 𝑦 = 𝑚𝑥 + 𝑏 equation and write your answer. Answer: 𝑦 = 6𝑥 + 17 The point-slope form of an equation of a line with slope m and containing the point (𝑥1 , 𝑦1 ) is: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) How to find an equation of a line given the slope and a point. Step 1. Step 2. Step 3. Step 4. Identify the slope. Identify the point. Substitute the values into the point-slope form, 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) Write the equation in slope-intercept form. For Example: Use the point slope formula to find the equation of a line with 1 𝑚 = 2 𝑎𝑛𝑑 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (4, −3). Write your answer in slope-intercept form. 1 Step 1. Slope: 𝑚 = 2 Step 2. Point (4,-3) 1 Step 3. Plug in x1 = 4, y1 = -3 and 𝑚 = into the equation 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 2 1 𝑦 − (−3) = (𝑥 − 4) 2 Step 4. Solve for y to get the equation written in slope intercept form. 𝑦+3= 1 𝑥−2 2 −3 −3 1 𝑦= 𝑥−5 2 Answer: 1 𝑦 = 2𝑥 − 5 (ℎ𝑖𝑛𝑡: 1 1 −4 −4 (−4) = × = = −2) 2 2 1 2 How to find an equation (written in 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form) of a line given two points. Step 1. Find the slope using the given points. 𝑚 = 𝑦2 −𝑦1 𝑥2 −𝑥1 Step 2. Choose one point. Step 3. Substitute the values into the point-slope form or slope-intercept form. Step 4. Write the equation in slope-intercept form. For Example: Use the point slope formula to find the equation of a line passing through the points(5, −2) 𝑎𝑛𝑑 (4,1). Write your answer in slope-intercept form. Step 1. Slope: 𝑚 = 1−(−2) 4−5 3 = −1 = −3 (𝑚 = −3) Step 2. Choose a point. You may choose either. Both points will give the same final answer. I choose the point (4,1) as I like to avoid working with negative numbers. Step 3. Can be done like the first example, or like the second example. You should pick the method that seems easier to you. In general, students make less mistakes using the second method (pointslope form) I would pick the point-slope method, unless you really do not like it. Step 3: performed is performed two ways below. Solved using the slope-intercept equation. Solved using the point slope formula Step 1. Slope: m = 3 Step 2. Point (4, 1) Step 3. Let x = 4, y = 1 and m = 3 in the equation y = mx + b and solve for b. Step 1. Slope: 𝑚 = 3 Step 2. Point (4,1) Step 3. Plug in x1 = 4, y1 = 1 and 𝑚 = 3 into the equation 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 1 = 3(4) + 𝑏 1 = 12 + 𝑏 −12 − 12 −11 = 𝑏 Step 4. Replace m with 3 and b with -11 in the y = mx + b equation and write your answer. Answer: 𝑦 = 3𝑥 − 11 𝑦 − 1 = 3(𝑥 − 4) Step 4. Solve for y to get the equation written in slope intercept form. 𝑦 − 1 = 3𝑥 − 12 +1 +1 𝑦 = 3𝑥 − 11 𝐴𝑛𝑠𝑤𝑒𝑟: 𝑦 = 3𝑥 − 11 How to find an equation (written in 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form) of a line parallel to a given line. Step 1. Find the slope of the given line. Step 2. Find the slope of the parallel line. Step 3. Identify the point. Step 4. Substitute the values into the point-slope or slope-intercept form. Step 5. Write the equation in slope-intercept form For example: Find the slope-intercept form of the equation of a line passing through the point (2,3) that is parallel to the line 𝑦 = 5𝑥 − 1 Step 1. The given line has a slope of m = 5 (the line is written in slope-intercept form and the number in front of the x is the slope.) Step 2. The slope of the parallel line is m = 5 (Parallel lines have equal slopes.) Step 3. Identify the point (2, 3) Step 4. I will use the point slope formula, but you can use the alternate method and find the answer using the slope-intercept formula. let 𝑚 = 5, 𝑥1 = 2 𝑎𝑛𝑑 𝑦1 = 3 and plug into the equation 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 𝑦 − 3 = 5(𝑥 − 2) Step 5. 𝑆𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦 𝑦 − 3 = 5𝑥 − 10 +3 +3 𝑦 = 5𝑥 + 7 Answer: 𝑦 = 5𝑥 + 7 Section 3.3: Equation of a line Chapter 3: Lines How to find an equation of a line (written in 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form) perpendicular to a given line. Step 1. Find the slope of the given line. Step 2. Find the slope of the perpendicular line. Step 3. Identify the point. Step 4. Substitute the values into the point-slope or slope-intercept form. Step 5. Write the equation in slope-intercept form. For example: Find the slope-intercept form of the equation of a line passing through the point (−6,7) 2 that is perpendicular to the line 𝑦 = 𝑥 + 9 3 2 Step 1. The given line has a slope of m = (the line is written in 𝑠𝑙𝑜𝑝𝑒 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 form and the 3 number in front of the 𝑥 is the slope.) −3 Step 2. The slope of the perpendicular line is m = (perpendicular lines have slopes that are 2 negative reciprocals.) Step 3. Identify the point (-6, 7) Step 4. I will use the point slope formula, but you can use the alternate method and find the answer using the slope-intercept formula. −3 let m = , 𝑥1 = −6 𝑎𝑛𝑑 𝑦1 = 7 and plug into the equation 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 2 −3 𝑦 − 7 = ( 2 ) (𝑥 − (−6)) Step 5. Write in slope-intercept form 𝑦−7= 𝑦−7= −3 𝑥−9 2 +7 +7 −3 𝑦= 𝑥−2 2 Answer: −3 𝑦 = 2 𝑥−2 −3 (𝑥 + 6) 2 (ℎ𝑖𝑛𝑡: −3 −3 6 −18 ×6= × = = −9 2 2 1 2