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Finding the Equation of a Line

Finding the Equation of a Line. To find the equation of a line you need to know the slope and y-intercept!. Instructions. There are 12 practice problems in this packet.

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Finding the Equation of a Line

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  1. Finding the Equation of a Line To find the equation of a line you need to know the slope and y-intercept!

  2. Instructions • There are 12 practice problems in this packet. • Record the answers to each problem on a sheet of notebook paper, along with any work that may be necessary to arrive at the answers. • Check each answer before advancing to the next problem. If you get one wrong, rework it until you can get the right answer; this is a study tool after all. • Turn in your paper for an easy 100! • You will not be penalized if you do not turn in your paper; this review is to help you, not hurt you! However, if you do not take it seriously, you will only hurt yourself!

  3. Slope-Intercept Form The slope-intercept form of a linear equation is… y = mx + b Where “m” is the slope and “b” is the y-intercept.

  4. Writing the equation of a line when given the slope and y-intercept: • Write the equation of a line with a slope of ½ and a y-intercept of -6: • Answer: y = ½ x – 6 • Write the equation of a line with the following characteristics: b = 2; m = -5 • Answer: y = -5x + 2

  5. Finding the Equation when given the Slope & a Point on the Line • In the equation y = mx + b, replace y, m, & x with the values given. • Solve for b. • Now that you know the slope and y-intercept, write the equation. Example: Find the equation of a line that passes through the point (8, -4) and has a slope of -2. -4 = (-2)8 + b -4 = -16 + b (Add 16 to both sides) 12 = b y = -2x + 12

  6. Practice Problem #1 Which function represents the line that contains the point (2, 12) and has a slope of -3? A. f(x) = -3x + 6 B. f(x) = -3x + 18 C. f(x) = -3x + 34 D. f(x) = -3x + 38 Check Answer 

  7. Practice Problem #2 • Which graph best represents the line that has a slope of and contains the point (4, -3)? A. C. B. D. Check Answer 

  8. FINDING THE SLOPE If you are given two points, the first step in any of the methods is to find the slope from the two given points, using the slope formula.

  9. Example 1: A line passes through the two points (-4, 1) and (2, 5). Find the equation of this line. • First, find the slope: • Next, use the slope & one of the points to solve for the y-intercept, b. • Now that you know the slope & y-intercept, write the equation: Or Use [STAT] Edit 

  10. Using the Graphing Calculator… • [STAT] Edit • Enter the x-values of the two points in List 1 • Enter the y-values of the two points in List 2

  11. To find the equation of the line… • [STAT] CALC • 4: LinReg • [ENTER] • Write the equation on your paper. • Convert the decimal numbers to fractions use the [MATH] button; see next slide .

  12. Converting Decimals to Fractions… • Enter .66666666666 (all the way across the screen) • [MATH] [ENTER] [ENTER] • Enter 3.666666666666 • [MATH] [ENTER] [ENTER] Write the Equation:

  13. Practice Problem #3 What is the y-intercept of the function shown in the graph? A. -24 B. -21 C. -18 D. -9 (Hint: Find the equation of the line using the two points given; graph it on your calculator; view the table to find the value of y when x = 0. (10, 17) (6, 3) Check Answer 

  14. Practice Problem #4 Which of the following describes the line containing the points (0, 4) and (3, -2)? • y = -2x + 4 • y = ½ x + 6 • y = 2x + 4 • y = - ½ x + 6 Check Answer 

  15. Standard Form of a Linear Equation The standard form of a linear equation is… Ax + By = C Where A is the coefficient of x, B is the coefficient of y, and C is the constant.

  16. Converting from Slope-Intercept Form to Standard Form • To convert from y = mx + b to Ax + By = C, move the term that includes x to the left side of the equation. • When writing linear equations in standard form, there are two things you must remember: 1) The leading coefficient – which is the coefficient of x – must be positive; and 2) No fractions are allowed! To get rid of the fractions, multiply the entire equation by the least common denominator (LCD).

  17. Examples y = - ½ x + 6 ½ x + y = 6 (Add ½ x to both sides.) 2( ½ x + y = 6) Multiply everything x 2) x + 2y = 12 • Y = ⅓x – ½ • -⅓x + y = - ½ (Subtract ⅓x from both sides) • -6(-⅓x + y = - ½ ) (Multiply everything x -6, the LCM of 3 & 2; you multiply by -6 instead of 6 in order to cancel the leading negative.) • 2x – 6y = 3 Converting from slope-intercept form to standard form is sometimes necessary because, on TAKS, they often write the answer choices in standard form.

  18. Practice Problem #5 Which equation best represents the line shown in the graph? A) 7x + 4y = 35 B) 4x – 7y = 35 C) 4x + 7y = -35 D) 7x – 4y = -35 (7, -1) (0, -5) Check Answer 

  19. Additional Practice Problems • #6) Convert this equation to standard form: y = ⅜x – 4 • #7) Convert this equation to standard form: y = -⅔x + ⅜ Check Answer  Check Answer 

  20. Converting from Standard Form to Slope-Intercept Form • This is a very important skill because you cannot graph linear functions on your calculator unless the equation is written in slope-intercept form. • When converting from standard form to slope-intercept form, your objective is to get y by itself. • Example: 3x – 5y = 15 • -5y = -3x + 15 (Subtract 3x from both sides) • Y = x – 3 (Divide everything by -5)

  21. Practice Problem #8 At what point does the line 3x + 5y = 7 intersect the x-axis? Hint: Since the x-intercept is the point where the value of y = 0, replace y with 0 and solve for x. Check Answer 

  22. Writing Equations from Tables of Data… • When you are asked to find the equation that matches a table of data, enter the data into your calculator using [STAT] EDIT and perform a linear regression. • Example, find the equation that matches this table of data. Click here for the answer.

  23. To match a Table to a Given Equation… • Enter the equation in the Y= screen of your calculator. • View the resulting table and compare it to the answer choices.

  24. Practice Problem #9 Which table best represents the function y = 2x − 6? A. B. C. Check Answer 

  25. Practice Problem #10 Matt is a speed skater. His coach recorded the following data during a timed practice period. If Matt continues to skate at the rate shown in the table, what is the approximate distance in meters he will skate in 20 seconds? A. 222 m B. 175 m C. 150 m D. 278 m Check Answer 

  26. Practice Problem #11 A math club decided to buy T-shirts for its members.  A clothing company quoted the following prices for the T-shirts. Which equation best describes the relationship between the total cost c, and the number of T-shirts, s? • c = 6.75s • c = 7.00s • c = 2s – 20 • c = 15 + 6s Check Answer 

  27. Practice Problem #12 Which of the following tables contains values for an equation that has a slope of 4? A. B. C. D. End Show Check Answer 

  28. Answer to Example Table: y = x + 2  Back

  29. Answer to Problem #1 B. y = -3x + 18 In the equation y = mx + b, replace y, m, & x with the values given & solve for b, as shown. Write the equation in slope-intercept form.  Back

  30. Answer to Problem #2: D. In the equation y = mx + b, replace y, m, & x with the values given & solve for b, as shown. Write the equation in slope-intercept form; enter it in Y= and hit [GRAPH].  Back

  31. Answer to Problem #3: C. -18 Enter the two points in the [STAT] EDIT screen, with the x-values going in L1 and the y-values in L2. Perform a linear regression by using [STAT] CALC 4 The value given for b is the y-intercept.  Back

  32. Answer to Problem #4: A. y = -2x + 4 Enter the two points in the [STAT] EDIT screen, with the x-values going in L1 and the y-values in L2. Perform a linear regression by using [STAT] CALC 4 to find the equation.  Back

  33. Answer to Problem #5 B. 4x – 7y = 35 Convert the equation to standard form in order to find the correct answer: The y-intercept is -5. The slope is , which you can find by beginning at the y-intercept & counting up 4 & right 7 until you are at another identifiable point on the line. Once you know the slope & y-intercept, you can write the equation:  Back

  34. Answer to Problem #6  Back

  35. Answer to Problem #7  Back

  36. Answer to Problem #8 C.  Back

  37. Answer to Problem #9 C. Enter the equation into the Y= screen of the calculator, view the resulting table and compare it to the answer choices. Be careful! Every ordered pair must match up; compare ALL of them, not just one or two.  Back

  38. Answer to Problem #10: A. 222 m Enter the points in the [STAT] EDIT screen, with the x-values going in L1 and the y-values in L2. Perform a linear regression by using [STAT] CALC 4 to find the equation. Enter the equation into Y=. View the table; find the y-coordinate when x = 20.  Back

  39. Answer to Problem #11: D. c = 15 + 6s Enter the points in the [STAT] EDIT screen, with the x-values going in L1 and the y-values in L2. Perform a linear regression by using [STAT] CALC 4 to find the equation.  Back

  40. Answer to Problem #12 B. • Enter each table – one at a time – in the [STAT] EDIT screen & perform a linear regression. • Look for the table which has a slope of 4, which is table B, as indicated below. This is the slope.  Back

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