Finding equations of lines


 Shauna Sims
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1 Finding equations of lines A very typical question for a student in a math class will be to find the equation of a line. This worksheet will provide several examples of how to complete this task. Find the equation of the line that goes through the point (2, 5) and has a slope of 3. STEP 1: Get the slope of the line The slope is already given to us and is equal to 3 STEP2: Write the equation of the line using y = mx + b or y y 1 = m(x In this example, I will use both.  Usingy = mx + b, let m=3 and x=2 and y=5. All we need to do is solve for b. To solve for b, plug in m, x, and y into the equation. 5=3*2+b 5=6 + b 1=b  Using y y 1 = m(x, let m=3 andx 1 = 2 and y 1 = 5. y 5 = 3(x 2) STEP 3: Simplify and solve the equation for y  Using y = mx + b. All we do here is replace m and b in the equation y = 3x 1  Using y y 1 = m(x. I will distribute the 3 and solve for y y 5 = 3x 6 y = 3x 1
2 As you can see, any of the two equations gives us the same answer, y = 3x 1 Find the equation of the line that goes through the points (1,5) and (3,4). STEP 1: Find the slope of the line In this example we are not given a slope, so we will need to find it. I will use the formula m = y 2 y 1. We use this formula when we are given two points on the x 2 x 1 line. I will let x 1 = 1, y 1 = 5, x 2 = 3 and y 2 = 4 m = 4 5 = 4 5 = 1 3 ( 1) STEP2: Write the equation of the line using y = mx + b or y y 1 = m(x I will use y y 1 = m(x (it s just personal preference!! Please remember that you can use y = mx + b). I will let m = 1, x 4 1 = 1, and y 1 = 5 Note you could have picked: x 1 = 3 and y 1 = 4 as well. y 5 = 1 (x ( 1)) 4 y 5 = 1 (x + 1) 4 STEP 3: Simplify and solve the equation for y. y 5 = 1 4 x 1 4 y = 1 4 x Find the equation of the line that is parallel to 3x 5y = 7 and goes through the point (1,3).
3 STEP 1: Find the slope of the line. Just like the example above, we are not given a slope. How do we get a slope? First, parallel lines have the same slope. Which means that the line we are looking for has the same slope as 3x 5y = 7. But, how do we find the slope of this line? We have to get the line into y intercept form or y = mx + b form if we do this we can get the slope. 3x 5y = 7 3x 3x 5y = 7 3x 5y = 3x + 7 5y 5 = 3x y = 3 5 x 7 5 (Subtrtact 3x from both sides) (Result after you subtract 3x) (Dividing both sides by 5) (Result after you divide by 5) We have our equation in this form y = mx + b form, so we can go ahead and get our slope which is m = 3 5 STEP 2: Write the equation of the line using y = mx + b or y y 1 = m(x I will use y y 1 = m(x. I will let m = 3 5, x 1 = 1, and y 1 = 3 y ( 3) = 3 (x 1) 5 y + 3 = 3 (x 1) 5 STEP 3: Simplify and solve the equation for y. y + 3 = 3 5 x 3 5 y = 3 5 x 18 5
4 Find the equation of the line that is perpendicular to 2x 6y = 4 and passes through the point (4, 4). STEP 1: Find the slope of the line. To get the slope of the line we have to use the fact that the slopes of perpendicular lines are negative reciprocals of each other. For example, if the slope of one line is 1 the slope of the line perpendicular to this line will have a slope of 2 = 2. An 2 1 easy way to remember this is to flip the fraction and change the sign. We have to get the line into y intercept form or y = mx + b form if we do this we can get the slope 2x 6y = 4 2x 2x 6y = 4 2x 6y = 2x + 4 6y 6 = 2x y = 1 3 x 2 3 (Subtract 2x from both sides) (Result after subtracting 2x from both sides.) (Divide both sides by 6) (Result after dividing by 6) We have our equation in this form y = mx + b form, so we can go ahead and get the slope of this line which is 1. Because our lines are perpendicular the slope of 3 our line is going to be m = 3 1 = 3. STEP 2: Write the equation of the line using y = mx + b or y y 1 = m(x I will use y y 1 = m(x. I will let m = 3, x 1 = 4, and y 1 = 4 y 4 = 3(x 4)
5 STEP 3: Simplify and solve the equation for y. y 4 = 3x + 12 y = 3x + 16 The next problem deals with finding the equation of the line with no slope. Similar situations work if the slope is zero. Find the equation of the line that goes through the point (1,9) and is parallel to x = 4. STEP 1: Find the slope With this problem, most of the work will be done here. The line is parallel to x = 4. Lines where x = any number are vertical lines. The lines that are parallel to vertical lines are vertical as well, so our line has to have the form x =number. Vertical lines have undefined slope so there is no slope. Step 2: Form the equation Our equation is x = number, because the line has to goes thorugh (1,9) the equation of the line is x = 1 so we are done. NOTE: When finding equations of lines you will sometimes you will be asked to find the equation of a line and then getting it into standard form Ax + By = C form, where A, B, and C are integers (no fractions or decimals) This next example illustrates how to do this with one the problems above. Find the equation of the line that is parallel to 3x 5y = 7 and goes through the point (1,3). Write the equation in standard form.
6 We know that the solution to this equation is y = 3 x 18. We know will get this 5 5 equation in standard form. 5 y = x (Multiply both sides by 5, the LCD) 5y = 3x 18 (Result after multiplying both sides by 5) 5y 3x = 3x 3x 18 (Subtract 3x from both sides) 5y 3x = 18 or 3x + 5y = 18 (Result after subtracting 3x from both sides) Hence our final answer is 3x + 5y = 18
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