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Finding the equation of a line. POINT-SLOPE FORM y – y1 = m ( x – x1 ) Bell Work Using the point-slope form, find the equation of the line with the given conditions. Express your answer in y=mx + b. 1. 2. 3. 4. 5. Having a slope of 3 and passing through the point ( 4, 5 ). Having a slope of -2 and passing through the point ( - 1, 6 ) m = 8 , ( 7, 4 ) m = -10 , ( 6, -8 ) m=5,(9,3) Take Note If the given is the slope and a point on the line, we can use the point-slope form which is. y – y1 = m ( x – x1 ) What if the given slope is a fraction? Try this! Find the equation of the line having a slope of ¾ and passing through ( 2, 5 ). Solution Given : m = ¾ and passing through ( 2,5 ) y – y1 = m ( x – x1 ) y–5=¾(x–2) y – 5 = ¾ x – 6/4 y = ¾ x – 6/4 + 5 y = ¾ x + 7/2 Another solution m = ¾ and passing through ( 2, 5 ) y – y1 = m ( x – x1 ) y–5=¾(x–2) 4(y–5)=3(x–2) 4y – 20 = 3x – 6 4y = 3x – 6 + 20 4y = 3x + 14 4 4 4 y = ¾ x + 7/ 2 Try this! Find the equation of the line having the slope of ½ and passing through ( - 3, 4 ) Try this! Find the equation of the line having a slope of 2/5 and passing through ( - 3, - 6 ) Try this! Find the equation of the line having a slope of -2/7 and passing through ( - 1, - 4 ) Think – Pair - Share Find the equation of the line given the slope and a point. 1. 2. 3. 4. 5. m = 2/3 ( 4, 7 ) m = ¼ ( 3, 1) m = 3/5 ( -2, 4) m = - 1/8 ( 3, -2) m=½ ( 0, -23 ) Finding Slopes and y - intercepts from Equations y = mx + b Objectives: Change a linear equation to the form y = mx + b. Determine the slope of a line and the y-intercept from the given equations. Thoughts to Ponder To determine the slope and y – intercept from a given equation, solve the equation for y in terms of x and express the resulting equation in the form y = mx + b. The coefficient of the xterm is the slope (m) and the constant term b represents the y-intercept. Example: Find the slope of the line and yintercept of 8x + y = 10. Solution: Given: 8x + y = 10 y = mx + b y = -8x + 10 Therefore the slope is -8 and the yintercept is 10. Example Find the slope and y-intercept in the line 2x + 2y = 12 Solution: 2x + 2y = 12 2y = -2x + 12 2 2 2 y = -x + 6 Slope = -1 : y-intercept = 6 Try this! Change 3y + 15 = 3x in y-form. Try this! Change 2y + 6x = 30 in y-form. Board Drill Change the following equations in the form y = mx + b. 10 x + y = 3 2x – 2y = 6 10x + 5y = 25 14 x- 7y = -12 2y – 4 = 35 – x Seatwork Change the following equations in the form y = mx + b, then determine the slope and yintercept. Given 1. 9x + y = 18 2. 6x + 2y = 20 3. 3y + 5x = 2y + 2 4. 3x + 6y = -21 5. 2x + 3y = 15 Y-form slope Y-intercept Standard Form of Linear Equations Ax + By = C Mathematical Concepts Standard This Form Ax + By = C is one of the two forms of a linear equation. The letters A, B, and C represent numbers. The numbers may not be fractions. Examples 1. Change the equation y = -2x + 5 in Standard Form Given: y = -2x + 5 Solution: Just put all the variable terms on the left side of the equation. Note: x – term must comes first Answer: 2x + y = 5 Examples 2. Change the equation y = 3x - 8 in Standard Form Given: y = 3x - 8 Solution: Just put all the variable terms on the left side of the equation. Note: the value of a should always be positive -3x + y = -8 -3x + y = -8 Multiply by -1 3x – y = 8 Examples 3. Change the equation y = -2/3x - 7 in Standard Form Given: y = -2/3 x - 7 Solution: Just put all the variable terms on the left side of the equation. Note: there must be no fractions in A, B, or C 2/3 x + y = -7 2/3 x + y = -7 Multiply by 3 2x + 3y = -21 Board Drill Write the following in standard form. y = 6x - 9 Board Drill Write the following in standard form. 3x – 5 = 2y Board Drill Write the following in standard form. 7 = 2x – 3y Board Drill Write the following in standard form. y = -3/4x -10 Board Drill Write the following in standard form. y = ½ x -7 Seatwork Write the following equations in standard form. y = -8x + 5 2. y = 10x + 2 3. 3x – 6 = 7y 4. y = 1/3 x – 5 5. y = -4/9 x + 3 1. Drills Group A: -3/5 x – 7y = 11 Group B y = 8x - 13 Group C 5 + 4x = y Drills Group A: -4/7 x – 2y = 3 Group B y = 9x + 2 Group C y = -2x - 7 Drills Group A: -3/8 x – y = 7 Group B y = -1/2 x + 9 Group C y = 3x - 4 Did you know that?...... Using the Standard Form of Linear Equation we can also determine the value of the slope and yintercept using the formula. SLOPE = − Y-INTERCEPT = Change the equation in standard form then find the values of a, b, and c 3x = 5y + 2 S. F 3x – 5y = 2 A=3 B = -5 C=2 Change the equation in standard form then find the values of a, b, and c y = 3x – 7 -3x + y = -7 Multiply by -1 3x – y = 7 Therefore: A = 3 B = -1 C=7 Change the equation in standard form then find the values of a, b, and c 3/4x = y – 7 ¾ x – y = -7 4 ( ¾ x – y = -7) 3x – 4y = -28 Therefore: A=3 B = -4 C = -28 Think-Pair-Share Complete the table below. Given 1. y = -4x + 5 2. y = 2x -7 3. 2x = 5 – 9y 4. y = 2/3x - 5 5. 7y = -4/9 x - 3 S.F A B C m b