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2.5 Writing Linear Equations Algebra 2 Mrs. Spitz Fall 2006 Objectives: Write the slope-intercept form of an equation given the slope and a point or two points, Write the standard form of an equation given the slope and a point, or two points and Write an equation of a line that is parallel or perpendicular to the graph of a given equation. Assignment pp. 76-78 #6-41 Introduction In Lesson 2.2, you learned if a function can be written in the form y = mx + b, then it is a linear equation. But what numbers do m and b represent? Finding Slope-Intercept form Look at the graph. The line passes through points A(0, b) and C(x, y). Notice that b is the y intercept of AC. Suppose you need to find the slope of AC. Now solve the equation for y. y b m x0 y b m x Finding Slope-Intercept form y b m x mx y b m x b y y m x b Multiply each side by x. Add b to each side Symmetric property of Equality You may recognize this as the form of a linear function. When an equation is written in the form y = mx + b, it is in slope-intercept form. 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. Notes If you are given the slope and y-intercept of a line, you can find an equation of the line by substituting values of m and b into the slopeintercept form of the equation. Then the equation can be written in standard form. For example, if you know the slope of a line is ⅔ and the y-intercept is 5, an equation of the line is 2 y x6 3 In standard form, 2x – 3y = -18 Ex. 1: Find the slope-intercept form of the equation of the line that has a slope of ¾ and passes through (8, 2). You know the slope and the x and y values of one point on the graph. Substitute for m, x and y in the slope-intercept form. y m x b 3 2 ( )(8) b 4 2 6b 4b Slope-Intercept form The equation in slopeintercept form is Substitute values y = ¾x - 4 Multiply Subtract 6 from both sides Standard Form Remember that the stand form of the equation of a line is Ax + By = C. Suppose we write this general equation in slopeintercept form. Ax By C By Ax C A C y x B B The slope is y-intercept is A and the B C , for B ≠ 0. B This can be used to write an equation in standard form when you are given the information you usually use to find the slopeintercept form. Subtract Ax from each side. Divide each side by B. Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1). First use the two given points to find the slope of the line. m y 2 y1 x2 x1 1 5 3 ( 2) 4 m 5 m If the slope is -4/5, then –A/B = (4/5). Thus A = 4 and B = 5 Substitute these values into the standard form. The resulting equation is 4x + 5y = C. Since one of the points on the line is (3, 1), you can substitute these values into the equation to find C. Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1). Ax By C Standard form of a linear equation. 4x 5 y C Substitute values for A and B. 4(3) 5(1) C Substitute values for x and y. 12 5 C Simplify 17 C Addition property of Equality Ex. 3: Application problem The atmospheric pressure at sea level is 14.7 pounds per square inch. As divers go deeper into the ocean, the pressure increases. Use the chart to write an equation in slopeintercept form that approximates this relationship. Then find the pressure at 30,000 feet below sea level. Depth Pressure (in feet) (lbs/in2) Sea level (0) 14.7 600 269 1200 536 3000 1338 7200 3208 18,000 8019 How? Let x represent the ocean dept and y represent the pressure. Use your calculator and a pair of points to find the slope of the line. 269 536 m 0.445 600 1200 You may want to use another set of points to confirm your slope. 3208 1338 m 0.445238095 7200 3000 Next? The slope of the line is approximately 0.445. The y- intercept corresponds to the ocean depth of 0 (at sea level). So the y-intercept is 14.7. An equation that approximates the pressure at certain ocean depths is y= 0.445x + 14.7 The equation that is derived may differ based on the set of points used to determine the slope. Use your calculator again to approximate pressure at 30,000. 0.445 x 30,000 + 14.7 = 13364.7 The result is 13,364.7 pounds per square inch. Ex. 4: Write an equation of the line that passes through (4, 6) and is parallel to the line whose equation is: y = 2/3x + 5. Parallel lines have the same slope, so the slope of both lines is 2/3. Use the slopeintercept form and the point (4, 6) to find the equation. y m x b 2 6 ( )(4) b 3 8 6 b 3 18 8 b 3 3 10 b 3 Slope-Intercept form Substitute values for x, y and m Multiply 2/3 by 4 Common denominator Subtract 8/3 from 18/3 to get the y-intercept. The y-intercept is 10/3. An equation of the line is y 2 10 x 3 3 What’s the standard form? 2 10 y x 3 3 A2 B3 C 10 Ax By C 2 x 3 y 10 2 10 y x 3 3 Ex. 5: Write an equation in standard form for the line that passes through (4, 6) and is perpendicular to the line whose equation is y=2/3x + 5. 3 m 2 A3 B2 The slope of the given line is 2/3. Since the product of this slope and the slope of a perpendicular line is -1, the slope of a perpendicular line is -3/2. You can use the slope -3/2 and the point (4,6) to write the equation in standard form. Ax By C Standard Form 3(4) 2(6) C Substitute values for A, B and x and y 12 12 C 24 C 3x 2 y 24 Multiply values Addition property of equality Substitute into the standard form. Ex. 2: Find the standard form of the equation that passes through (-2, 5) and (3, 1). First use the two given points to find the slope of the line. y m x b 3 2 ( )(8) b 4 2 6b 4b Slope-Intercept form The equation in slopeintercept form is Substitute values y = ¾x - 4 Multiply Subtract 6 from both sides