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## Finding the Equation of a Line

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**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. • 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!**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.**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**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**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 **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 **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.**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 **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**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 .**Converting Decimals to Fractions…**• Enter .66666666666 (all the way across the screen) • [MATH] [ENTER] [ENTER] • Enter 3.666666666666 • [MATH] [ENTER] [ENTER] Write the Equation:**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 **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 **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.**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).**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.**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 **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 **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)**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 **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.**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.**Practice Problem #9**Which table best represents the function y = 2x − 6? A. B. C. Check Answer **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 **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 **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 **Answer to Example Table:**y = x + 2 Back**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**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**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**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**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**Answer to Problem #6** Back**Answer to Problem #7** Back**Answer to Problem #8**C. Back**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**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**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**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