Question 1
Solve the following simultaneous equations for variables x and y: 2x = 28 --- Eq 1 2y + 3x = -128 --- Eq 2
Solution: For this particular question, in order to use substitution method, we first need to calculate value of one variable. Looking at the equations above, this can easily be done with Eq 1 as it has only one variable.
Objective 1: Calculate value of x from Eq 1:
Given from Eq 1 that 2x = 28 Divide by 2 on both sides x = (28/ ? )
Simplify x =? --- A
Objective 2: Plug-in value of x from A in Eq 2
Given from Eq 2 that 2y + 3x = -128 Substitute value of x from A above 2y + 3(? ) = -128 Simplify 2y + ? = -128 2y = - 128 - ? 2y = - ? y = - ? Answer: x =? ; y = -?
100
Question 2
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 7y = 49 --- Eq 1 13x - 2y = 12 --- Eq 2 Answer: x = y =
Question 3
Solve the following simultaneous equations for variables x and y: 81x = 243 --- Eq 1
45x + 21y = 9 --- Eq 2
Solution: For this particular question, in order to use substitution method, we first need to calculate value of one variable. Looking at the equations above, this can easily be done with Eq 1 as it has only one variable.
Objective 1: Calculate value of x from Eq 1
Given from Eq 1 that 81x = 243
Divide by 81 on both sides x = (243/? )
Simplify x = ? --- A
Objective 2: Plug-in value of x from A in Eq 2
Given from Eq 2 that 45x + 21y = 9
Substitute value of x from A above 45(? ) + 21y = 9
Simplify ? + 21y = 9 21y = 9 - ? 21y = - ? y = -? Answer: x = ? ; y = -?
102
Question 4
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 72y = 216 --- Eq 1 32x - 54y = 126 --- Eq 2 Answer: x = y =
Question 5
Solve the following simultaneous equations for variables x and y: 12x + 3x = 45 --- Eq 1
4x - 5y = 62 --- Eq 2
Solution: For this particular question, in order to solve for variables x and y, we first need to combine the like variables. We can see that Eq 1 has all x variables. Therefore, they can be combined.
Objective 1: Combine x-terms in Eq 1
Given from Eq 1 that 12x + 3x = 45
Add x-terms as ? x = 45
Divide by 15 on both sides gives x = (45/? )
Simplify x = ? --- A
Objective 2: Plug-in value of x from A in Eq 2
Given from Eq 2 that 4x - 5y = 62 Substitute value of x from A above 4(? ) - 5y = 62 Simplify ? - 5y = 62 -5y = 62 - ? -5y = ? Divide by -5 y = -? Answer: x = ? ; y = -?
104
Question 6
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 5y - 12y = 21 --- Eq 1 5y - 23x = 77 --- Eq 2 Answer: x = y =
Question 7
Solve the following simultaneous equations for variables x and y: 2x + 13x = 75 --- Eq 1
-x + 2y = 5 --- Eq 2
Soution: For this particular question, in order to solve for variables x and y, we first need to combine the like variables. We can see that Eq 1 has all x-terms. Therefore, they can be combined.
Objective1: Combine x-terms in Eq 1
Given from Eq 1 that 2x + 13x = 75
Adding all x-terms ? x = 75
Dividing by the coefficient of x on both sides x = (75/? ) Simplify x =? --- A
Objecitve 2: Plug-in value of x in Eq 2
Given from Eq 2 that -x + 2y = 5
Substitute value of x from A above - ? + 2y = 5
Simplify 2y = 5 + ?
2y = ?
Divide by 2 on both sides y =? Answer: x = ? ; y =?
106
Question 8
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 2x - 70x = 204 --- Eq 1 3x + 2y = 19 --- Eq 2 Answer: x = y =
Question 9
Solve the following simultaneous equations for variables u and v: 13u = 4(13 - 26v) --- Eq 1 8u - v = -98 --- Eq 2
Solution: In order to solve this system of equations, we need to simplify Eq 1. Once simplified, we may use substitution method to find u and v.
Objective 1: Simplify Eq 1
Given from Eq 1 that 13u = 4(13 - 26v)
Multiply all terms in parenthesis with 4 13u = 4(? ) - 4(? v)
Simplify 13u = 52- 104v --- Eq 1.1
Objective 2: In order to use substitution, one variable needs to be expressed in terms of the other variable. This seems possible for Eq 1.1.
Given from Eq 1.1 that 13u = 52 - 104v Divide by 13 on both sides u = (52/13) - (104v/13) Simplify u = ? - ? v --- A
Objecitve 3: Substituting expression for u from A above in Eq 2 Given from Eq 2 that 8u - v = -98
Plug-in u from A above 8( ? - ? v) - v = -98 Simplify 8( ? ) - 8( ? v) - v = -98 ? - ? v - v = -98
- ? v = -98 - ?
-65v = - ?
v = ? --- B
Objecitve 4: Plug-in value of v from B in A
Given from A that u = 4 - 8v Substitute value of v from B above u = 4 - 8( ? ) Simplify u = 4 - ?
u = - ? Answer: u = -? ; v = ?
108
Question 10
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 8x = -2(28 + 8y) --- Eq 1 y = 6x + 9 --- Eq 2 Answer: x = y =
Question 11
Solve the following simultaneous equations for variables x and y: 4a + 3b = 12 - 5b --- Eq 1 3a + 2b = 17 --- Eq 2
Solution: As you can see, in Eq 1, the b-terms appear on both sides. Therefore, in order to use substitution to solve for variables a and b, we first need to combine like terms.
Objective 1: Combine like terms in Eq 1
Given from Eq 1 that 4a + 3b = 12 - 5b Combine b-terms 4a + 3b + ? b = 12
Simplify 4a + ? b = 12 --- Eq 1.1
Objective 2: In order to use substitution method to solve for variables a and b, we need to express one variable in terms of other variable. Glancing at Eq 1 and Eq 1.1, this is possible with Eq 1.1 as it has all compatible numbers.
Given from Eq 1.1 that 4a + ? b = 12 Express a in terms of b 4a = 12 - ? b
Divide by 4 on both sides (4a/4) = (12/4) - ( ? b/4) Simplify a = ? - ? b --- A
Objective 2: Plug-in a from A in Eq 2
Given from Eq 2 that 3a + 2b = 17 Plug-in a from a above 3( ? - ? b) +2b = 17 Solve 3( ? ) - 3( ? b) + 2b = 17 Simplify ? - ? b + 2b = 17 Move the constant term to the right side - ? b = 17 - ?
Solve -4b = ?
b = - ? --- B
Objective 3: Plug-in value of b in A
Given from A that a = ? - ? b Substitute value of b from B above a = 3 - 2(- ? ) Simplify a = 3 + ?
a =? Answer: a = ? ; b = -?
110
Question 12
Using paper and pencil provided to you, solve the following simultaneous equations for x and y: 6a - 7b = 21 - a --- Eq 1 10a + 4b = 170 --- Eq 2 Answer: a = b =