7.6 Properties of Linear Equations pg 442-447 outcome R9 1. Determine the number of solutions of each linear system. a) + = 3 x y = y –2x – –2 b) 4x + 6y = –10 3y = 5 –2x – 2x 4y = c) – –1 3x 6y = 2 – Solution Write each equation in slope-intercept form to identify the slope and of each line. y-intercept a) + = 3 x y = y –2x – –2 For equation : + = 3 Subtract from each side. x y x = + 3 y –x The slope is and the is 3. y-intercept –1 For equation : = Add 2x to each side. y –2x – –2 = 2x 2 Multiply by to for y. –y – –1 = + 2 y –2x The slope is and the is 2. y-intercept –2 Because the slopes are different, the lines intersect at exactly one point and the linear system has one solution. b) 4x + 6y = –10 3y = 5 –2x – For equation : 4x + 6y = Subtract 4x from each side. –10 6y = 10 Divide by 6 to solve for y. –4x – = – = y The slope is and the is . y-intercept For equation : 3y = 5 Add 2x to each side. –2x – = 2x + 5 Divide each side by to solve for –3y –3 y. = y Equation is the same as equation , so the is and the is . y-intercept (Solution continues.) The forms of both equations are the same, so the lines are coincident and the linear system has infinite solutions. 2x 4y = c) – –1 3x 6y = 2 – For equation :
7.6 Properties of Linear Equations pg 442-447 outcome R9 1. Determine the number of solutions of each linear system. a) x + y = 3 –2x – y = –2 b) 4x + 6y = –10 –2x – 3y = 5 c) 2x – 4y = –1 3x – 6y = 2 Solution Write each equation in slope-intercept form to identify the slope and y-intercept of each line. a) x + y = 3 –2x – y = –2 For equation : x + y = 3 Subtract x from each side. y = –x + 3 The slope is –1 and the y-intercept is 3. For equation : –2x – y = –2 Add 2x to each side. –y = 2x – 2 Multiply by –1 to for y. y = –2x + 2 The slope is –2 and the y-intercept is 2. Because the slopes are different, the lines intersect at exactly one point and the linear system has one solution. b) 4x + 6y = –10 –2x – 3y = 5 For equation : 4x + 6y = –10 Subtract 4x from each side. 6y = –4x – 10 Divide by 6 to solve for y. = – y = The slope is and the y-intercept is . For equation : –2x – 3y = 5 Add 2x to each side. –3y = 2x + 5 Divide each side by –3 to solve for y. y = Equation is the same as equation , so the is and the y-intercept is . (Solution continues.) The forms of both equations are the same, so the lines are coincident and the linear system has infinite solutions. c) 2x – 4y = –1 3x – 6y = 2 For equation :
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