Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics

The Slope and Equation An equation of a line is given by y = mx + c where m is a slope and c is a constant. For example, y = 4x -2 where 4 is a slope and -2 is a constant.

Examples 1. Find an equation of the line containing the given points (-8,-3) and (10,15). Solution: m = (15 + 3)/(10 + 8) = 18/18 = 1 y = mx + c -3 = 1 (-8) + c c = 8 -3 = 5 y = x + 5

2. Find the slope of the line 4x – 3y = 6 We need to write this in the form y = mx + c 3y = 4x -6 y = 4/3 x -2 The slope m = 4/3

Parallel and perpendicular lines Two lines are parallel if their slopes are equal. For example, y = 2x + 1 and y = 2x -3 are parallel lines because the slope = 2. Two lines are perpendicular if their slopes are negative reciprocals of each other. For example, y = 2x +5 and y = -1/2 x + 1 are perpendicular to each other because the negative reciprocal of 2 is -1/2.

examples 1. Determine whether the lines are parallel or perpendicular or neither. a) 3x – 4y = -15, and 3x – 4y = 6 -4y = -3x -15, and -4y = -3x + 6 y = ¾ x + 15/4, and y = ¾ x – 6/4 The lines have the same slope ¾, so they are parallel. b) 5x – y = 3, and x + 5y = 3 -y = -5x + 3, and 5y = -x + 3 y = 5x -3, and y = -1/5 x + 3/5 The slopes are negative reciprocal 5 and -1/5 so they are perpendicular to each other.

Skill Practice 1. Find the slope of the line containing the points (3,4) and (-1,1). 2. Find the slope of the line with x-intercept 4 and y- intercept Find an equation of the line containing the point (-4, 2) with slope -3/2. 4. Find an equation of the line with y-intercept -4 and slope 3.

Skill Practice 5. Find an equation to the line in the following figure: X y

Skill Practice 6. Find an equation of the line containing the point (2, -3) and (-1, 2). 7. Determine whether the lines 3x - 7y = 28 and 7x + 3y =3 are parallel or perpendicular or neither. 8. Find an equation of the line containing (2,3) and perpendicular to the line x –y = Find an equation of the line parallel to the line x=6 containing the point (-3,2). 10. Determine whether the line 2x -3y =1 and -4x +6y = 5 are parallel or perpendicular or neither.

Solution to Skill Practice 1. ¾ 2. points are (4,0) and (0,-5) m = (-5 -0) / (0-4) = 5/4 3. y = mx+ c y = -3/2x – 4 4. y = 3x points (0,2) and (3,0). So y = -2/3 x + 2

Solution to Skill Practice 6. y = -5/3 x + 1/3 7. Perpendicular to each other because slopes are negative reciprocal 3/7 and -7/3. 8. y = x -5 and negative reciprocal of 1 is -1. So the equation y = -x The line x = 6 is vertical, so the line we want is also vertical. The vertical line that goes through (-3,2) is x = The slopes are the same. So they are parallel to each other.

Applications of Lines and Slopes There are many applications of linear equations to business and science. These are called mathematical models. Example: A hospital paid $52.50 for water in January when they used 15,000 gallons and $77.50 in May when they used 25,000 gallons. Find an eqaution that gives the amount of the water bill in terms of gallons of water used. Points are (15000,52.50) and (25000,77.50). So y = x+15.

Examples The dosage of medication given to an adult patient is 20 mg plus 2 mg per pound. Find an equation that gives the amount of medication (in mg) per pound of weight. Solution: 20 mg is the y intercept. The slope m = increase in medication/increase in weight m = 2/1. So y = 2x + 20

Skill Practice 1. A sales representative earns a monthly base salary plus a commission on sales. Her pay this month will be $2000 on sales of $ last month her pay was $2720 on sales of $ Find an equation that gives her monthly pay in terms of her sales level. 2. The temperature scales Fahrenheit and Celsius are linearly related. Water freezes at 0 0 C and 32 0 F. Water boils at F and C. Find an equation that gives degrees Celsius in terms of degrees Fahrenheit.

Skill Practice 3. A sales manager believes that each $100 spent on television advertising results in an increase of 45 units sold. If sales were 8250 units sold when $3600 was spent on television advertising, find an equation that gives the sales level in terms of the amount spent on advertising.

Solution to Skill Practice 1. y = 3/25 x C = 5/9 F – 160/9 3. y = 9/20 x