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1. True or False: The ordered pair (-4, 5) is a solution of the following system of equations: 3X - 4Y= -32 -7X + 10Y = 78 We can substitute x and y values to the equation to verify if the pair is solution or not. We know x = -4 and y = 5, then to the first equation 3 * (-4) – 4 * 5 = -32 -7 * (-4) + 10 *5 = 28 + 50 = 78 Therefore, the pair (-4, 5) satisfies both equations. It is a solution of the system. 2. Which one is addition method and which one is substitution method? (20x -3y = 14) and (4x + 10y = -8) or (x = 3y + 14) and (4x + 10y = -8) The second one (x = 3y + 14) and (4x + 10y = -8) is substitution method. In the first equation, x is expressed as a function of y, we then can substitute it for x in the second equation to solve y. (20x -3y = 14) and (4x + 10y = -8) The above system is addition method. We can multiply -5 to the second equation, then add it to the first equation to cancel out x terms and solve for y. 3. What is the slope of the line passing A(5,0), and B(9,40). y yB y A 40 0 40 10 The slope is m x xB x A 95 4 So the slope of the line passing through the above two points is 10. 4. Which one is: Parallel, Perpendicular, or neither. x = -3y + 4 y = -3x -3 y= 3x + 2 When two lines are parallel, their slopes are equal. When two lines are perpendicular, the product of their slopes is equal to -1. We first write all line equations in the slope-intercept form: 1 3 x 3 y 4 3 y x 4 y x 3 4 So the slope is -1/3 The slope of line y = -3x – 3 is -3 The slope of the line y = 3x + 2 is 3 We know that (-1/3) * 3 = -1 Therefore, the lines x = -3y + 4 and y = 3x + 2 are perpendicular. 5. Solve the system by any method. x= 7/4 y + 3 4x - 7y =12 We can solve it by substitution method because the first equation is in the form of x as polynomial of y. Substituting it to the second equation 7 4 y 3 7 y 12 4 7 y 12 7 y 12 00 The above equation is always true. So the system has infinite numbers of solutions given 4x – 7y = 12. It is call underdetermined system. 6. What is the slope of the line passing through the points A(5,2) and B(0,0). y y A yB 2 0 2 The slope is m x x A xB 5 0 5 7. Determine whether the graphs of the pair of equations are parallel, perpendicular or neither. 9x + 3y = 9 3x - 9y = 1 Write the equations in the slope-intercept form y = mx + b. 9 x 3 y 9 3 y 9 x 9 y 3x 1 , so the slope m1 = -3. 3x 9 y 1 9 y 3x 1 y 1 1 x , sol the slope is m2 = 1/3. 3 9 1 m1m2 3* 1 3 Therefore, the two lines are perpendicular. 8. Write an equation of the line that passes through the given point and is parallel to the given line. Write the answer in slope-intercept form. (0,6), 4y - 12x = 2 4 y 12 x 2 4 y 12 x 2 y 3 x 1 2 So the slope of the line 4y – 12x = 2 is 3. The unknown line is parallel to the above line, then its slope is 3 too. The line passes through (0, 6), so its y-intercept b = 6. In the slope-intercept form y = mx + b m = 3, b = 6. So the line equation is y = 3x + 6. 9. If the following system is to be solved using the addition method, by what constant should one of the equations be multiplied if the x terms are to drop out? -6x - 2y= -8 -12x - 3y= 3 The coefficient the x in the first equation is -6. The coefficient of x in the second equation is -12. so we can multiply -2 to the first equation to obtain -6x * (-2) – 2y * (-2) = -8 *(-2) 12x + 4y = 16 Adding the above equation to the second equation 4y – 3y = 19 y = 19 Substituting y = 19 to the first equation -6x – 2 * 19 = -8 -6x = 30 x = -5 so the solution is x = -5, y = 19