Math 0070
Module 4 Goal 10.2
Simultaneous Equations
Write your answers in the grey boxes and then click on the blue box to check your answer.
Solve the following systems of linear equations by using the addition/subtraction method. Checkyour answer before you click on the blue box. (a) x – 3y = 6
2x + 3y = 3
Answer: Add the equations. 3x = 9 x = 9/3 x = 3 Substitute x = 3 into x - 3y = 6 3 - 3y = 6 -3y = 6 - 3 -3y = 3 y = 3/-3 y = -1 (3, -1)
(b)2x – 3y = 4
2x + y = -4
Answer: Subtract the equations. -4y = 8 y = 8/-4 y = -2 Substitute y = -2 into 2x - 3y = 4 2x - 3(-2) = 4 2x + 6 = 4 2x = 4 - 6 2x = -2 x = -2/2 x = -1 (-1. -2)
(c) x – 2y = 1
2x – 3y = 4
Answer: Multiply the first equation by 2 and leave the second equation alone. 2x - 4y = 2 2x - 3y = 4 Subtract the equations. -y = -2 y = -2/-1 y = 2 Substitute y = 2 into x - 2y = 1 x - 2(2) = 1 x - 4 = 1 x = 1 + 4 x = 5 (5, 2)
(d) x – 2y = 9
3x – 6y = 17
Answer: Multiply the first equation by 3 and leave the second equation alone. 3x - 6y = 27 3x - 6y = 17 Subtract the equations. 0 = 10 no solution (inconsistent)
(e)3x – 6y = 12
2x – 4y = 8
Answer: Multiply the first equation by 2. Multiply the second equation by 3. 6x -12y = 24 6x - 12y = 24 Subtract the equations. 0 = 0 infinite number of solutions (dependent)
Solve the following systems of linear equations by using the substitution method. Check your answer before you click on the blue box. (a)y = 2x + 10
2x + y = -2
Answer: Substitute 2x + 10 from the first equation into y in the second equation. 2x + 2x + 10 = -2 4x = -2 - 10 4x = -12 x = -12/4 x = -3 Substitute x = -3 into y = 2x + 10 y = 2(-3) + 10 y = -6 + 10 y = 4 (-3, 4)
(b)x = 2y + 1
2x – 3y = 4
Answer: Substitute 2y + 1 into x in the second equation. 2(2y + 1) - 3y = 4 4y + 2 - 3y = 4 4y - 3y = 4 - 2 y = 2 Substitute y = 2 into x = 2y + 1 x = 2(2) + 1 x = 4 + 1 x = 5 (5, 2)
(c)9x – 3y = 15
y = 3x – 5
Answer: Substitute 3x - 5 from the second equation into y in the first equation. 9x - 3(3x - 5) = 15 9x - 9x + 15 = 15 15 = 15 infinite number of solutions (dependent)
(d)4x – 2y = 3
y = 2x – 2
Answer: Substitute 2x - 2 from the second equation into y in the first equation. 4x - 2(2x - 2) = 3 4x - 4x + 4 = 3 4 = 3 no solution (inconsistent)
(e)3y = 5 – 2x
x + 3y = 7
Answer: Substitute 5 - 2x from the first equation into 3y in the second equation. x + 5 - 2x = 7 x - 2x = 7 - 5 -x = 2 x = 2/-1 x = -2 Substitute x = -2 into 3y = 5 - 2x 3y = 5 - 2(-2) 3y = 5 + 4 3y = 9 y = 9/3 y = 3 (-2, 3)
(f)3x + 2y = 7
2y = 9x + 11
Answer: Substitute 9x + 11 from the second equation into 2y in the first equation. 3x + 9x + 11 = 7 12x = 7 - 11 12x = -4 x = -4/12 x = -1/3 Substitute x = -1/3 into 2y = 9x + 11 2y = 9(-1/3) + 11 2y = -3 + 11 2y = 8 y = 8/2 y = 4 (-1/3, 4)
Solve the following systems of linear equations by using either the addition/subtraction method or by using the substitution method. The answers show the solution by one of the methods, but the same solution obtained by using any method is acceptable. Check your answer before you click on the blue box. (a)2x – y = 5
6x + 2y = -5
Answer: Multiply the first equation by 2 and leave the second equation alone. 4x - 2y = 10 6x + 2y = -5 Add the equations. 10x = 5 x = 5/10 x = 0.5 or x = 1/2 Substitute x = 0.5 into 2x - y = 5 2(0.5) - y = 5 1 - y = 5 -y = 5 - 1 -y = 4 y = 4/-1 y = -4 (0.5, -4) or (1/2, -4)
(b)3x – 6y = 15
4x – 8y = 20
Answer: Multiply the first equation by 4 and multiply the second equation by 3. 12x - 24y = 60 12x - 24y = 60 Subtract the equations. 0 = 0 infinite number of solutions (dependent)
(c)2x + 3y = 8
x = 2y – 3
Answer: Substitute 2y - 3 from the second equation into x in the first equation. 2(2y - 3) + 3y = 8 4y - 6 + 3y = 8 4y + 3y = 8 + 6 7y = 14 y = 14/7 y = 2 Substitute y = 2 into x = 2y - 3 x = 2(2) - 3 x = 4 - 3 x = 1 (1, 2)
(d) x + 2y = 7
2x + 4y = 9
Answer: Isolate x in the first equation. x = 7 - 2y Substitute 7 - 2y from the first equation into x in the second equation. 2(7 - 2y) + 4y = 9 14 - 4y + 4y = 9 14 = 9 no solution (inconsistent)
(e)0.3x – 0.7y = 0.4
0.2x + 0.5y = -0.7
Answer: Multiply each equation by 10 to change the decimals to whole numbers. 3x - 7y = 4 2x + 5y = -7 Now, multiply the new first equation by 2 and the new second equation by 3. 6x - 14y = 8 6x + 15y = -21 Subtract the equations. -29y = 29 y = 29/-29 y = -1 Substitute y = -1 into 0.3x - 0.7y = 0.4 0.3x - 0.7(-1) = 0.4 0.3x + 0.7 = 0.4 0.3x = 0.4 - 0.7 0.3x = -0.3 x = -0.3/0.3 x = -1 (-1, -1)
M070 FWC gp 10.2
