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Linear Relations and Functions B-4 Writing Linear Equations ACT WARM-UP Ten thousand tires are being stored in a warehouse. Two percent of the tires are not usable. What is the ratio of usable tires to not usable tires? A) 1 98 B) 1 50 C) 1 49 D) 49 1 E) 50 1 If 2% are not usable, then 98% are usable. 49 Therefore 98:2 is simplified to D) . 1 Objectives Write an equation of a line given the slope and a point on the line Write an equation of a line parallel or perpendicular to a given line Essential Question How do you write an equation of a line given the coordinates of two points on a line? Slope-Intercept Form of a Linear Equation The slope-intercept form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. y = mx + b slope y-intercept If you are given the slope and y-intercept of a line, you can quickly find an equation of the line by substituting the values of m and b into the slope-intercept form. For example, if you know that the slope of a line is – 3 and the y-intercept is 4, the equation of the line is y = – 3x + 4, or, in standard form, 3x + y = 4. In previous courses you have also learned that the slope-intercept form can be used to find an equation of a line given the slope and the coordinates of any point on the line. Write an equation in slope-intercept form for the line that has a slope of and passes through (5, –2). Slope-intercept form – Simplify. Add 3 to each side. Answer: The y-intercept is 1. So, the equation in slope-intercept form is More often in algebra, when you are given slope and a point or two points on a line most students find it difficult and tedious to solve for the y-intercept (b) in the slope-intercept form. An easier method to use given slope and a point or given two coordinates on a line is the point-slope form. Point-Slope Form of a Linear Equation The point-slope form of the equation of a line is y – y1 = m(x – x1), where (x1,y1) are the coordinates of a point on the line and m is the slope of the line. y – y1 = m(x – x1) slope coordinates of point on line Write an equation of the line that passes through (5, 4) and has a slope of −3. Solution Because you know the slope and a point on the line, use point-slope form to write an equation of the line Let 𝑥1 , 𝑦1 = 5, 4 and m = −3. 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 𝑦 − 4 = −3 𝑥 − 5 𝑦 − 4 = −3𝑥 + 15 𝑦 = −3𝑥 + 19 Use point-slope form Substitute for m, 𝑥1 , 𝑎𝑛𝑑 𝑦1 Distributive property Write in slope-intercept form. Write an equation in slope-intercept form for the line that has a slope of Answer: and passes through (–3, –1). Multiple-Choice Test Item What is an equation of the line through (2, –3) and (–3, 7)? A B C D Read the Test Item You are given the coordinates of two points on the line. Notice that the answer choices are in slope-intercept form. Solve the Test Item First, find the slope of the line. Slope formula Simplify. The slope is –2. That eliminates choices B) and C) . Then use the point-slope formula to find an equation. Point-slope form you can use either point for . Distributive Property Subtract 3 from each side. Answer: D Multiple-Choice Test Item What is an equation of the line through (2, 5) and (–1, 3)? A C B D Answer: C When changes in real-world situations occur at a linear rate, a linear equation can be used as a model for describing the situation. Sales As a part-time salesperson, Jean Stock is paid a daily salary plus commission. When her sales are $100, she makes $58. When her sales are $300, she makes $78. Write a linear equation to model this situation. Let x be her sales and let y be the amount of money she makes. Use the points (100, 58) and (300, 78) to make a graph to represent the situation. Slope formula Simplify. Now use the slope and either of the given points with the point-slope form to write the equation. Point-slope form Distributive Property Add 58 to each side. Answer: The slope-intercept form of the equation is What are Ms. Stock’s daily salary and commission rate? The y-intercept of the line is 48. The y-intercept represents the money Jean would make if she had no sales. Thus, $48 is her daily salary. The slope of the line is 0.1. Since the slope is the coefficient of x, which is her sales, she makes 10% commission. Answer: Ms. Stock’s daily salary is $48, and she makes a 10% commission. How much would Jean make in a day if her sales were $500? Find the value of y when Use the equation you found in Example 3a. Replace x with 500. Simplify. Answer: She would make $98 if her sales were $500. Sales The student council is selling coupon books to raise money for the Humane Society. If the group sells 200 books, they will receive $150 dollars. If they sell 500 books, they will make $375. a. Write a linear equation to model this situation. Answer: b. Find the percentage of the proceeds that the student council receives. Answer: 75% c. If they sold 1000 books, how much money would they receive to donate to the Humane Society? Answer: $750 The slope-intercept and point-slope forms can be used to find equations of lines that are parallel or perpendicular. Write an equation for the line that passes through (3, –2) and is perpendicular to the line whose equation is The slope of the given line is –5. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular line is Use the point-slope form and the ordered pair (3, –2) to write the equation. Point-slope form Distributive Property Add –2 to each side. Answer: An equation of the line is Write an equation for the line that passes through (3, 5) and is perpendicular to the line whose equation is Answer: Essential Question How do you write an equation of a line given the coordinates of two points on a line? Find the slope using the slope formula and use the pointslope form, y – y1 = m(x – x1), to substitute the slope for m and one of the given points for (x1,y1). Math Humor Student: I’ll just draw a quick line by hand and guess the slope. Teacher: No, no. It’s point-slope form, not point-sloppy form.