Simultaneous Equations

Question 1

Solve the following system of simultaneous equations

x+ 5y = 34

(1)

x - 5y = - 6 .

(2)

Which of the following is the solution for $x ?$

Not correct. Choice (a) is false.

Try again, this is the value of $y .$

Not correct. Choice (b) is false.

Try again. You need to add the equations together to eliminate $y.$

Your answer is correct.

Solving by elimination we add the two equations and get $2x = 28$ and hence $x = 14.$
This then gives $y = 4$ since by substitution into (1) we have $34--14- 20 y = 5 = 5 = 4$ and we check (2) and see that $14 - 5(4) = - 6.$ So we have that $x = 14$ and $y = 4.$

Not correct. Choice (d) is false.

Try again. You need to add the equations together to eliminate $y .$

Question 2

Solve the system of equations below.

2m + 3n = 14

(1)

4m - 5n = 17.

(2)

Which of the following is the solution for $m .$

Not correct. Choice (a) is false.

Try again, this is the value of n.

Not correct. Choice (b) is false.

Try again, check the sign.

Not correct. Choice (c) is false.

Try again, multiply (1) by 2 to eliminate $m$ and solve for $n$ then ﬁnd $m .$

Your answer is correct.

Solving by elimination we multiply (1) by 2 to give

4n + 6n = 28

(3)

Subtract (2) from (3) to give $n = 11.$ Substitute $n = 11$ into (1) and

19 2m + 33 = 14 ⇒ 2m = - 19 ⇒ m = --2

Check in (2) and $- 38+ 55 = 17.$ Hence $n = 11$ and $m = 9.5.$

Question 3

Consider the simultaneous equations below

2x + 5y = 16

(1)

3x - 2y = 5.

(2)

Which of the following are the correct steps to eliminate $x$ from the equations?

Not correct. Choice (a) is false.

Try again, this eliminates $y .$

Not correct. Choice (b) is false.

Try again, you would need to subtract the equations to eliminate $x .$

Not correct. Choice (c) is false.

Try again, this gives $- 5x + 16y = 17.$

Your answer is correct.

These steps give

6x+ 15y = 48

- 6x+ 4y = - 10.

and

19y = 38 ⇒ y = 2.

Question 4

Which of the following is the solution for a to the simultaneous equations

2a +3b = - 5

(1)

3a - 2b = 12

(2)

Not correct. Choice (a) is false.

Try again. To eliminate $b$ we multiply (1) by 2 and (2) by 3 and add the equations.

Not correct. Choice (b) is false.

Try again. To eliminate $b$ we multiply (1) by 2 and (2) by 3 and add the equations.

Not correct. Choice (c) is false.

Try again. To eliminate $b$ we multiply (1) by 2 and (2) by 3 and add the equations.

Your answer is correct.

Solving by elimination we multiply (1) by 2 and (2) by 3.

4a+ 6b = - 10

(3)

9a - 6b = 36

(4)

Add equations (3) and (4) and get $13a = 26$ and hence $a = 2.$
Substitute $a = 2$ into equation (1) and get $4+ 3b = - 5$ so $b = - 3 .$
Check in equation (2) and 3(2)-2(-3)=12 and we have $a = 2$ and $b = - 3.$

Question 5

Consider the system of simultaneous equations below

5(2w - z) = 7w + 1

(1)

3(3w + z) = 5(w- z + 12) .

(2)

Which of the following is the correct simpliﬁcation of the problem and solution for $w$ to the system?

Not correct. Choice (a) is false.

Try again, you the correct equations but you have solved for $z .$

Not correct. Choice (b) is false.

Try again, you have not multiplied 12 by 5. These are the wrong equations but it is the correct solution for $z$ to the wrong equations.

Your answer is correct.

Expanding the equations we have

10w - 5z = 7w + 1

9w+ 3z = 5w - 5z + 60

Collecting like terms we have

3w- 5z = 1

4w + 8z = 60

Dividing the last equation by 4 we have the required equations

3w- 5z = 1

(3)

w+ 2z = 15

(4)

Multiplying (4) by 3 we have

3w + 6z = 45

(5)

Subtract (3) from (5) and we have $11z = 44 ⇒ z = 4.$ Substituting into (4) we have $w + 8 = 15 ⇒ w = 7$
Check in (1) and we have $3(3 × 7+ 4) = 5(7- 4+ 12) = 75 .$ Hence $w = 7.$

Not correct. Choice (d) is false.

Try again, you have not multiplied 12 by 5. These are the wrong equations but it is the correct solution to the wrong equations.


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