Section 1.5. Finding Linear Equations
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1 Section 1.5 Finding Linear Equations
2 Using Slope and a Point to Find an Equation of a Line Example Find an equation of a line that has slope m = 3 and contains the point (2, 5). Solution Substitute m = 3 into the slope-intercept form: y = 3 x+ b. Now we must find b Every point on the graph of an equation represents of that equation, we can substitute x = 2 and y = 5 Slide 2
3 Using Slope and a Point to Find an Equation of a Line Solution Continued 5 = 3(2) + b Substitute 2 for x and 5 for y. 5 = 6 + b Multiply. 5 6 = 6 + b 6 Subtract 6 from both sides. 1 = b Simplify. We now substitute 1 for b into y = 3x + b: y = 3x 1 Slide 3
4 Using Slope and a Point to Find an Equation of a Line Graphing Calculator We can use the TRACE on a graphing calculator to verify that the graph of y = 3x 1contains the point (2, 5). Slide 4
5 Using Two Points to Find an Equation of a Line Example Find an equation of a line that contains the points ( 2, 6) and (3, 4). Solution Find the slope of the line: m = = = = 2 3 ( 2) We have y = 2x+ b Line contains the point (3, 4) Substitute 3 for x and 4 for y Slide 5
6 Using Two Points to Find an Equation of a Line Example 4 = 2(3) + b Substitute 3 for x. 4 for y. 4 = 6 + b Multiply = 6 + b + 6 Add 6 to both sides. 2 = b Simplify. Substitute 2 for b into y = 2x + b: y = 2x + 2 Slide 6
7 Using Two Points to Find an Equation of a Line Graphing Calculator We can use the TRACE on a graphing calculator to verify that the graph of y = 2x + 2 contains the points ( 2, 6) and (3, 4). Slide 7
8 Finding a Linear Equation That Contains Two Given Points Guidelines To find the equation of a line that passes through two given points whose x-coordinates are different, y2 y1 1. Use the slope formula, m =, to find the x 2 x 1 slope of the line. 2. Substitute the m value you found in step 1 into the equation y = mx + b. Slide 8
9 Finding a Linear Equation That Contains Two Given Points Guidelines Continued 3. Substitute the coordinates of one of the given points into the equation you found in step 2, and solve for b. 4. Substitute the m value your found in step 1 and the b value you found in step 3 into the equation. y = mx + b. 5. Use a graphing calculator to check that the graph of your equation contains the two points. Slide 9
10 Using Two Points to Find an Equation of a Line Example Find an equation of a line that contains the points ( 3, 5) and (2, 1). Solution First we find the slope of the line: 1 ( 5) m + = = = 2 ( 3) We have y = x + b. 5 The line contains the point (2, 1) Substitute 2 for x and 1 for y: Slide 10
11 Using Two Points to Find an Equation of a Line Solution Continued 4 1 = (2) + b Substitute 2 for x. 1 for y = + b (2) = = ( 1) = b Multiply both sides by = 8 + 5b 5 = = = = 5b Subtract 8 from both sides. 13 = b Divide both sides by 5..5 Slide 11
12 Using Two Points to Find an Equation of a Line Solution Continued So, the equation is y = 4 x 5 Graphing Calculator 13.5 We can use the TRACE on a graphing calculator to verify that 4 13 the graph of y = x contains 5 5 the points ( 3, 5) and (2, 1). Slide 12
13 Finding an approximate Equation of a Line Example Find an approximate equation of a line that contains the points ( 6.81, 7.17) and ( 2.47, 4.65). Round the slope and the constant term to two decimal places. Solution First we find the slope of the line: m = = ( ) Slide 13
14 Finding an approximate Equation of a Line Solution Continued We have y = 0.58x + b. Since the line contains the point ( 6.81, 7.17), we substitute 6.81 for x and 7.17 for y: 7.17 = 0.58( 6.81) + b 7.17 = b = b b Sub for x, 7.17 for y. Multiply. Subtract from both sides. Combine like terms. Slide 14
15 Finding an approximate Equation of a Line Solution Continued So, the equation is y = 0.58x Graphing Calculator We can use the TRACE on a graphing calculator to verify that the graph of y = 0.58x comes very close to the points ( 6.81, 7.17) and ( 2.47, 4.65). Slide 15
16 Defining Point-Slope Form Method 2: Using Point- Slope Second method to find a linear equation of a line. Suppose that a nonvertical line has: Slope is m y-intercept is (x 1, y 1 ) (x, y) represents a different point on the line y y1 So, the slope is: = m x x 1 Slide 16
17 Defining Point-Slope Form Method 2: Using Point- Slope Given the slope, multiple both sides by x x 1 gives y y1 (x x 1 ) = m (x x 1 ) x x1 y y 1 = m (x x 1 ) We say that this linear equation is in point-slope form. Definition If a nonvertical line has slope m and contains the point (x 1, y 1 ), then an equation of the line is y y 1 = m (x x 1 ) Slide 17
18 Using Point-Slope Form to Find an Equation of a Line Example A line has slope m = 2 and contains the point (3, 8). Find the equation of the line Solution Method 2: Using Point- Slope Substituting x 1 = 3, y 1 = 8 and m = 2 into the equation y y 1 = m (x x 1 ). Slide 18
19 Using Point-Slope Form to Find an Equation of a Line Example Method 2: Using Point- Slope Use point-slope form to find an equation of the line that contains the points ( 5, 2) and (3, 1). Then write in slope-intercept form. Solution First find the slope of the line: m = = = ( ) Slide 19
20 Using Point-Slope Form to Find an Equation of a Line Solution Continued Method 2: Using Point- Slope Substituting x 1 = 3, y 1 = 1 and m = equation y y 1 = m (x x 1 ). 3 8 into the Slide 20
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