Question 1
Solve the following systems of equations for variables x and y:
24 x + 13y = 107 --- Eq 1 8x + 16y = 24 --- Eq 2
Solution: In order to use substitution method, we need to express one variable in terms of the other variable. Looking at the equations above, this can easily be done with Eq 2 as it has all compatible numbers.
Objective 1: Express x in terms of y in Eq 2:
Given that 8x + 16y = 24 --- Eq 2 Subtract 16y from both sides: 8x = 24 - 16y
Divide by 8 on both sides: x = (24/8) - (16y/8) Simplify: x = 3-2y --- A Objective 2: Plug-in the value of x from A in Eq 1 as
Given that 24 x + 13y = 107 Plug-in value of x from A above 24 (3 - 2y) + 13y = 107 Solve 72- 48y + 13y = 107 Simplify - 35y = 35 y = - 1 ---B Objective 3: Substitute value of y = -1 in A to get x as
Given from A that: x = 3 - 2y Substitute y = -1: x = 3 - 2(-1) Simplify: x = 3 + 2 x = 5 Answer: x = 5; y =-1
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Question 2 Instructions:
• Solve the following simultaneous equations for the missing variables.
• You are not required to show all steps (like the questions on the computer) • You may use your own method to solve these simultaneous equations. • Write answers in the space provided.
• You are not allowed to use any technological device to solve these equations. • Only when you have finished solving any specific question on paper, press the
forward arrow button on the computer screen to proceed to Question 3. • Please don't press the forward arrow button before or during solving any
question.
7x - 14y = 28 --- Eq 1 18x - 28y = 40 --- Eq 2
Question 3
Solve the following simultaneous equations for x and y:
4 (4x + 5y = 50) --- Eq 1 4x + 12y = 120 --- Eq 2 Solution: First, since Eq 1 has parenthesis, we need to simplify
Given that Eq 1 4(4x + 5y = 50) --- Eq 1 Multiply by 4 to all numbers inside parenthesis 16x + 20y = 200 --- Eq 1.1
In order to use substitution method, we need to express one variable terms of the other variable.
Looking at the equations 1.1 and 2 above, this can easily be done with Eq 2 as it has
all compatible numbers.
Objective 1: Express x in terms of y in Eq 2
Given that 4x + 12y = 120 Subtract 12y from both sides 4x = 120 - 12y Divide by 4 on both sides x = (120/4) - (12y/4)
Simplify x = 30 - 3y --- A
Objective 2: Plug-in value of x from A in Eq 1.1
Given that 16x + 20y = 200
Plug - in value of x from A above 16 (30- 3y) + 20y = 200 Multiply by 16 to the numbers in parentheses 480 - 48y + 20y = 200 Combine like terms -48y +20y = 200 - 480 -28y = - 280
Solve for y y = (-280/- 28)
y = 10 --- B Objective 3: Substitute value of y = ? in A to get x as
Given from A that x = 30 - 3y Substitute value of y from B above x = 30 – 3 (10) Simplify x = 30 - 30
x = 0 Answer: x = 0; y = 10
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Question 4 Instructions:
• Solve the following simultaneous equations for the missing variables.
• You are not required to show all steps (like the questions on the computer) • You may use your own method to solve these simultaneous equations. • Write answers in the space provided.
• You are not allowed to use any technological device to solve these equations. • Only when you have finished solving any specific question on paper, press the
forward arrow button on the computer screen to proceed to Question 3. • Please don't press the forward arrow button before or during solving any
question.
9x - 18y = 36 --- Eq 1 2(x + 4y = 40) --- Eq 2
Question 5 Solve the following simultaneous equations for x and y: 38x + 73y + 3y = 76 --- Eq 1 23x + 121y = 346 --- Eq 2
Solution: Since Eq 1 has two y terms that can be combined, we need to simplify
Given that 38x + 73y + 3y = 76 --- Eq 1 Combine y terms 38x + 76y = 76 --- Eq 1.1
In order to use substitution method, we need to express one variable terms of the other variable.
Looking at the equations 1.1 and 2 above, this can easily be done with Eq 1.1 as it has all
compatible numbers.
Objective 1: Express x in terms of y in Eq 1.1
Given that 38x + 76y = 76 Subtract 76y from both sides 38x = 76 - 76y
Divide by 38 on both sides x = (76/38) - (76y/38)
Simplify x = 2 - 2y --- A
Objective 2:Plug-in value of x from A in Eq 2
Given from A that x = 2 - 2y
Given from Eq 2 that 23x + 121y = 346 ---- Eq 2 Plug - in value of x from A in eq 2 23 (2 - 2y) + 121y = 346 Solve 46 - 46y + 121y = 346 Combine y terms 46 + 75y = 346 Subtract constant term from both sides 75y = 346 - 46 Simplify 75y = 300 Divide by the coefficient of y y = 300/75
y = 4 --- B Objective 3: Substitute value of y from B above in A to get x as
Given from A that x = 2 - 2y Substitute y = 4 x = 2 - 8 Solve x = - 6 Answer: x = -6; y = 4
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Question 6 Instructions:
• Solve the following simultaneous equations for the missing variables.
• You are not required to show all steps (like the questions on the computer) • You may use your own method to solve these simultaneous equations. • Write answers in the space provided.
• You are not allowed to use any technological device to solve these equations. • Only when you have finished solving any specific question on paper, press the
forward arrow button on the computer screen to proceed to Question 3. • Please don't press the forward arrow button before or during solving any
question.
72x - 2x - 30y = - 50 --- Eq 1 63x - 252y = 630 --- Eq 2
Question 7
Solve the following simultaneous equations for a, b and c: 2a = 12 --- Eq 1
2b - c = -13 --- Eq 2 3a + b = 13 --- Eq 3
Solution: Since we have three equations here, we need to find one equation from which the value of the variable can be found easily. That equation would be Eq 1.
Objective 1: Equate value of a variable
Given from Eq 1 that 2a = 12 Divide by 2 on both sides a = 12/ 2
Solve a = 6 --- A
Objective 2: Substitute value of a from A in Eq 3
Given from Eq 3 that 3a + b = 13 Substitute a A 3(6) + b = 13 Solve 18 + b = 13 b = 13 - 18
b = - 5 --- B
Objective 3: Substitute value of b from B in Eq 2
Given from Eq 2 that 2b - c = -13 Substitute b from B 2(-5) - c = -13 Solve - 10 - c = -13 -c = -13 + 10 -c = - 3 c = 3 --- C Answer: a = 6, b = - 5, c = 3
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Question 8 Instructions:
• Solve the following simultaneous equations for the missing variables.
• You are not required to show all steps (like the questions on the computer) • You may use your own method to solve these simultaneous equations. • Write answers in the space provided.
• You are not allowed to use any technological device to solve these equations. • Only when you have finished solving any specific question on paper, press the
forward arrow button on the computer screen to proceed to Question 3. • Please don't press the forward arrow button before or during solving any
question.
4b = 16 --- Eq 1 2a - c = 12 --- Eq 2
Question 9 Solve the following simultaneous equations for u and v: 8(u + v) = 24 --- Eq 1
4(u + 3v) = 52 --- Eq 2
Solution: Since both Eq 1 and Eq 2 have parenthesis, we need to simplify them before using substitution to solve.
Objective 1: Simplify Eq 1 and Eq 2
Given that 8(u + v) = 24 --- Eq 1 Mulyiply by 8 8u + 8v = 24 --- Eq 1.1 Given that 4(u + 3v) = 52 --- Eq 2 Multiply by 4 4u + 12v = 52 --- Eq 2.1
We will work with Eq 1.1 and Eq 2.1 as they are in simplified form. In order to use substitution method, we need to express one variable terms of the other variable. Looking at the equations 1.1 and 2.1 above, this can easily be done with Eq 1.1 as it has
all compatible numbers.
Objective 2: Express u in terms of v in Eq 1.1
Given from Eq 1.1 that 8u + 8v = 24 To get the u term all by itself, subtract the v term from both sides 8u = 24 - 8v Divide by coefficient of u on both sides u = (24/8) - (8v/8) Simplify u = 3 - v --- A Objective 3: Plug-in value of u from A in Eq 2.1
Given from A that u = 3 - v Given from Eq 2.1 that 4u + 3v = 52
Plug-in value of u from A 4(3 - v) + 3 v = 52 Simplify 12 - 4v + 12v = 52 Collect all v terms on left side and constant terms on the right side -4v + 12v = 52 - 12 Simplify 8v = 40
Divide by the coefficient of v on both side to solve for v v = 5 --- B
Objective 4: Plug-in value of v from B in A Given from A that u = 3 - v Substitute v from B u = 3 - 5
Solve u = - 2 --- C Answer: u = - 2, v = 5
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Question 10 Instructions:
• Solve the following simultaneous equations for the missing variables.
• You are not required to show all steps (like the questions on the computer) • You may use your own method to solve these simultaneous equations. • Write answers in the space provided.
• You are not allowed to use any technological device to solve these equations. • Only when you have finished solving any specific question on paper, press the
forward arrow button on the computer screen to proceed to Question 3. • Please don't press the forward arrow button before or during solving any
question.
10(u + 2v = 20) --- Eq 1 20(2u + 3v = 12) --- Eq 2
Question 11 Solve the following simultaneous equations for x, y and z:
6x + 12y - 42z = - 36 --- Eq 1 7x - 5y + 8z = 72 --- Eq 2 2x + 8y + 3z = -65 --- Eq
Solution: In order to use substitution method, we need to express one variable terms of other variables. Looking at the equations 1,2 and 3 above, this can easily be done with Eq 1 as it has all compatible numbers.
Objective 1: Express x in Eq 1 in terms of y and z Given from Eq 1 that 6x + 12y - 42z = - 36 Subtract 12y from both sides 6x - 42z = -36 - 12y Add 42z to both sides 6x = -36 - 12y + 42z
Divide by 6 on both sides x = (-36/6) - (12y/6) + (42z/6) Solve x = - 6 - 2y + 7z --- A Objective 2: Plug-in value of x from A in Eq 2
Given from Eq 2 that 7x - 5y + 8z = 72
Plug-in x from A 7(-6 - 2y + 7z) - 5y + 8z = 72 Multiply by 7 to all numbers in parenthesis - 42 - 14y + 49z - 5y + 8z = 72 Collect all y and z terms together - 14y - 5y + 49z + 8z - 42 = 72
Add 42 to both sides - 14y - 5y + 49z + 8z = 72 + 42
Simplify - 19y + 57z = 114 Objective 3:Get y term all by itself
Subtract z term from both sides -19y = 114 - 57z Divide by coefficient of y on both sides y = (114/-19) - (57/- 19)