In which quadrant or on which axis do the following points lie?
a. A (5, −1)
b. B (−2, 3)
c. C (3, 0)
d. D (4, −2)
e. E (2, 0)
f. F (0, 4)
In which quadrant or on which axis do the following points lie?
a. A (4, −1)
b. B (−5, 0)
c. C (−2, −7)
d. D (0, −3)
e. E (6, 6)
f. F (5, 4)
Plot the following points and join them in the order A, B, C, D. Identify the type of quadrilateral and find its area and perimeter.
a. A (6, −3)
b. B (6, −6)
c. C (−2, −6)
d. D (−2, −3)
Plot the following points and join them in the order P, Q, R, S. Identify the type of quadrilateral and find its area and perimeter.
a. P (–2, 4)
b. Q (–8, 4)
c. R (–8, –2)
d. S (–2, –2)
For Problems 5 to 10, graph the equations using a table of values with four points.
[latex] 4x - y = 2 [/latex]
[latex] 2x + 3y = 12 [/latex]
[latex] x + y - 4 = 0 [/latex]
[latex] x + 2y - 4 = 0 [/latex]
[latex] \displaystyle{y = \frac{1}{2}x + 2} [/latex]
[latex] \displaystyle{y = -\frac{1}{2}x - 2} [/latex]
For Problems 11 to 16, graph the equations using the x-intercept, y-intercept, and another point.
[latex] 3x - 4y = 12 [/latex]
[latex] x - 2y = -1 [/latex]
[latex] x - 2y - 6 = 0 [/latex]
[latex] 3x + y - 4 = 0 [/latex]
[latex] y = 4x [/latex]
[latex] x = 2y [/latex]
For Problems 17 to 22, graph the equations using the slope and y-intercept method.
[latex] y = 4x + 6 [/latex]
[latex] y = 5x + 4 [/latex]
[latex] 3x + 2y - 12 = 0 [/latex]
[latex] 2x + 3y + 6 = 0 [/latex]
[latex] \displaystyle{y = -\frac{3}{4}x - 1} [/latex]
[latex] \displaystyle{y = -\frac{1}{3}x - 1} [/latex]
For Problems 23 to 28, determine the equation of the line in slope-intercept form that passes through the following points:
(3, 2) and (7, 5)
(4, 6) and (2, 4)
(5, –4) and (–1, 4)
(0, –7) and (–6, –1)
(1, –2) and (4, 7)
(3, –4) and (–1, 4)
Write the equation of a line parallel to [latex] 3x - 4y = 12 [/latex] and that passes through the point P(−2, 3).
Write the equation of a line parallel to [latex] x - 3y = 9 [/latex] and that passes though the point P(2, –3).
Write the equation of a line perpendicular to [latex] 2y = x + 4 [/latex] and that passes through the point P(–2, 5).
Write the equation of a line perpendicular to [latex] 3x + 4y + 6 = 0 [/latex] and that passes through the point P(4, –1).
For Problems 33 to 38, solve the systems of equations by using the Graphical method.
[latex] y = -2x - 1 [/latex]
[latex] y = 3x - 11 [/latex]
[latex] y = 2x + 3 [/latex]
[latex] y = -2x - 1 [/latex]
[latex] 2x - 3y - 6 = 0 [/latex]
[latex] x + 2y - 10 = 0 [/latex]
[latex] 3x + 4y - 5 = 0 [/latex]
[latex] 2x - y + 4 = 0 [/latex]
[latex] 2y = x [/latex]
[latex] y = -x + 3 [/latex]
[latex] 3y = 2x [/latex]
[latex] y = -3x + 11 [/latex]
Paul wants to hire a plumbing company to do work on a condominium project. Company A charges $250 for the initial consultation and $40/hour for labour. Company B charges $200 for the initial consultation and $50/hour for labour.
Write an equation in slope-intercept form to represent the total fees charged by each plumbing company.
Sketch the graphs and label the lines.
Where do the lines intersect? What does this point represent?
What does the graph indicate regarding which company should be hired?
Paul also wants to hire an electrical company for the condominium project. Company A charges $500 for the initial consultation and $50/hour for labour. Company B charges $300 for the initial consultation and $60/hour for labour.
Write an equation in slope-intercept form to epresent the total fees charged by each electrical company.
Sketch the graphs and label the lines.
Where do the lines intersect? What does this point represent?
What does the graph indicate regarding which company should be hired?
For Problems 41 to 46, determine whether each of the systems of equations has one solution, no solution, or many solutions without graphing.
[latex] 3x - 2y + 13 = 0 [/latex]
[latex] 3x + y + 7 = 0 [/latex]
[latex] 4x + 6y - 14 = 0 [/latex]
[latex] 2x + 3y - 7 = 0 [/latex]
[latex] x - 3y + 2 = 0 [/latex]
[latex] 3x - 9y + 11 = 0 [/latex]
[latex] 15x + 3y = 10 [/latex]
[latex] 5x + y = -3 [/latex]
[latex] 2x - 4y = 6 [/latex]
[latex] x - 2y = 3 [/latex]
[latex] 3x - y + 2 = 0 [/latex]
[latex] 9x - 3y + 6 = 0 [/latex]
For Problems 47 to 52, solve the systems of equations by using the Substitution method.
[latex] x + 4y = 8 [/latex]
[latex] 2x + 5y = 13 [/latex]
[latex] x + y = 3 [/latex]
[latex] 2x - y = 12 [/latex]
[latex] x + 4y + 12 = 0 [/latex]
[latex] 9x - 2y - 32 = 0 [/latex]
[latex] x - y - 1 = 0 [/latex]
[latex] 2x + 3y - 12 = 0 [/latex]
[latex] 3x + 2y = 5 [/latex]
[latex] y = 2x - 1 [/latex]
[latex] 4x + 3y = 12 [/latex]
[latex] 9 - 3x = y [/latex]
For Problems 53 to 58, solve the systems of equations by using the Elimination method.
[latex] 8x + 7y = 23 [/latex]
[latex] 7x + 8y = 22 [/latex]
[latex] 2x + y = 8 [/latex]
[latex] 3x + 2y = 7 [/latex]
[latex] 9x - 2y = -32 [/latex]
[latex] x + 4y = -12 [/latex]
[latex] 5x - 2y + 3 = 0 [/latex]
[latex] 3x - 2y - 1 = 0 [/latex]
[latex] 4x + 3y = 12 [/latex]
[latex] 18 - 6x = 2y [/latex]
[latex] 2x + y + 2 = 0 [/latex]
[latex] 6x = 2y [/latex]
For Problems 59 to 64, solve the systems of equations by using either the Substitution method or the Elimination method.
[latex] 0.4x - 0.5y = -0.8 [/latex]
[latex] 0.3x - 0.2y = 0.1 [/latex]
[latex] 0.2x - 0.3y = -0.6 [/latex]
[latex] 0.5x + 0.2y = 2.3 [/latex]
[latex] \displaystyle{\frac{5x}{3} - \frac{5y}{2} = -5} [/latex]
[latex] \displaystyle{\frac{x}{3} - \frac{y}{4} = 2} [/latex]
[latex] \displaystyle{\frac{x}{4} + \frac{y}{2} = 2} [/latex]
[latex] \displaystyle{\frac{x}{6} + \frac{2y}{3} = \frac{4}{3}} [/latex]
[latex] (2x + 1) - 2(y + 7) = -1 [/latex]
[latex] 4(x + 5) + 3(y - 1) = 28 [/latex]
[latex] 2(3x + 2) + 5(2y + 7) = 13 [/latex]
[latex] 3(x + 1) -4(y - 1) = -15 [/latex]
Find the two numbers whose sum is 95 and the difference between the larger number and the smaller number is 35.
Find the two numbers whose sum is 84 and the difference between the larger number and the smaller number is 48.
300 tickets were sold for a theatrical performance. The tickets cost $28 for adults and $15 for kids. If $7,230 was collected, how many adults and children attended this play?
640 tickets were sold for a soccer game between Toronto FC and Liverpool FC. The tickets cost $35 for adults and $20 for students. If $16,250 was collected from sales, how many adults and students attended the game?
A fruit punch contains 30% orange juice, while another contains 15% orange juice. How many litres of each should be mixed to make 60 litres of punch that contains 21% orange juice?
One type of Antifreeze is 16% alcohol and a second type of Antifreeze is 9% alcohol. How many litres of each should be mixed to obtain 35 litres of Antifreeze that is 12% alcohol?
A canoeist paddled for four hours at a speed of 4 km/h with the current and it took ten hours for the return trip against the same current. Calculate the average speed of the canoe in still water.
A patrol boat travelled for five hours at a speed of 9 km/h against the current and it took three hours for the return trip with the same current. Calculate the average speed of the boat in still water.
Girija drove at an average speed of 60 km/h. One hour later, Aran drove from the same location in the same direction at an average speed of 100 km/h. How many hours did Girija drive before Aran catches up to her?
Ship A leaves the port at an average speed of 75 km/h. After two hours, Ship B leaves from the same port and heads in the same direction at an average speed of 90 km/h. How many hours later will Ship B pass Ship A?