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6-2: Writing Linear Equations in Point-Slope and Standard Forms

6-2: Writing Linear Equations in Point-Slope and Standard Forms. OBJECTIVE: You should be able to write linear equations in point-slope form and in standard form. We will be writing a lot of equations of lines in this class. In Algebra 2, you will write many more.

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6-2: Writing Linear Equations in Point-Slope and Standard Forms

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  1. 6-2: Writing Linear Equations in Point-Slope and Standard Forms OBJECTIVE: You should be able to write linear equations in point-slope form and in standard form. We will be writing a lot of equations of lines in this class. In Algebra 2, you will write many more. Something to keep in mind when finding these equations is that you need to know two things: (1) A POINT ON THE LINE (2) THE SLOPE OF THE LINE Once you know a point and the slope, you can plug these values into a formula and get the equation for the line.

  2. 6.2.1 POINT-SLOPE FORM OF A LINEAR EQUATION For a given point (x1, y1) on a non-vertical line having slope m, the point-slope form of a linear equation is as follows. y - y1 = m(x - x1) 6-2: Writing Linear Equations in Point-Slope and Standard Forms Remember the four general slope types you need to memorize: • positive slope - up and to the right • negative slope - down and to the right • zero slope - horizontal line Equation will always be “y = #”. • no slope - vertical line Equation will always be “x = #”. The first formula we use to write equations of lines is: Find a point and the slope, then plug those values into the formula above. You might need to simplify your answer in some cases.

  3. y - = (x - ) y - = (x - ) 6-2: Writing Linear Equations in Point-Slope and Standard Forms EXAMPLE 1: Write the point-slope form of an equation for a line that passes through (-3, 5) and has a slope of m (x1, y1) Label everything in the problem. y - y1 = m(x - x1) Plug the values into the formula. Negative-negative yields positive. 5 -3/4 -3 An equation of the line is: y - 5 = -3/4(x + 3) EXAMPLE 2: Write the point-slope form of an equation for a horizontal line that passes through (-6, 2) (x1, y1) Label everything in the problem. Horizontal lines have what slope? 0 This is your m. Plug the values into the formula. y - y1 = m(x - x1) 2 0 -6 y - 2 = 0 Zero times anything yields zero. Re-write by adding 2 to both sides. An equation of the line is: y = 2

  4. 6.2.2 STANDARD FORM OF A LINEAR EQUATION The standard form of a linear equation is Ax + By = C where A, B, and C are integers, A  0, and A and B are not both zero. 6-2: Writing Linear Equations in Point-Slope and Standard Forms You cannot use the Point-Slope Form for vertical lines. Remember that vertical lines always have the form “x = #”. This leads into writing linear equations in another form. You will have to rewrite equations into Standard Form.

  5. +5x +5x -20 -20 6-2: Writing Linear Equations in Point-Slope and Standard Forms EXAMPLE 3: Write in standard form. 4( ) ( )4 Make it easier on yourself. Get rid of the fraction by multiplying both sides by 4. The fours on the right cancel, leaving... 4(y + 5) = -5(x - 2) Distribute. Get the x’s and y’s on the same side by… adding 5x to both sides. 4y + 20 = -5x + 10 Put the x’s in front and make sure it is a positive number. If not positive, change all signs. 5x + 4y + 20 = 10 Get rid of the plain numbers on the left by… subtracting 20 from both sides. 5x + 4y = -10 The next example adds another step to take prior to plugging into the point-slope formula.

  6. 6-2: Writing Linear Equations in Point-Slope and Standard Forms Remember, you need to know (1) a point and (2) the slope. Sometimes, you are given only two points. You still need the slope. However, you already know a formula to find the slope if you have two points. When you are given only two lines, you first have to use the slope formula to find the slope. You then choose either point and pair it with the slope you found to plug into the point-slope formula.

  7. -x -x -18 -18 6-2: Writing Linear Equations in Point-Slope and Standard Forms EXAMPLE 4: Write the point-slope form and the standard form of the line that passes through (-8, 3) and (4, 5). (x1, y1) (x2, y2) Label everything. Plug into slope formula to find slope. Now that you know m, plug the values into the point-slope formula. y - y1 = m(x - x1) 6y - 18 = x + 8 Point-slope form (first half of answer) -x + 6y - 18 = 8 x - 6y + 18 = -8 Now for standard form. Multiply both sides by 6 to get rid of the fraction. Distribute and simplify. x - 6y = -26 Move the x’s by subtracting x. The coefficient of x is negative, so change all signs. Move the 18 by subtracting 18 to both sides. Standard form (second half of answer)

  8. 6-2: Writing Linear Equations in Point-Slope and Standard Forms HOMEWORK Page 336 #19 - 43 odd

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