Systems of Linear Equations — Challenge

Two conditions at once. The answer is the single pair of numbers that satisfies both.

What these problems cover

These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

If x+y=10 and x-y=4, what is x?

  1. 3
  2. 5
  3. 6
  4. 7
  5. 14
Worked answer

7

Adding the two equations eliminates y, because +y and -y cancel.

(x+y)+(x-y) = 10+4 gives 2x = 14, so x = 7.

Then y = 3, and both original equations check out.

Problem 2

If y=2x and x+y=9, what is x?

  1. 2
  2. 3
  3. 4
  4. 6
  5. 9
Worked answer

3

The first equation already gives y in terms of x, so substitute it into the second.

x+2x = 9 gives 3x = 9, so x = 3.

Then y = 6, and 3+6 = 9 as required.

Problem 3

If 2x+y=11 and x-y=1, what is y?

  1. 1
  2. 2
  3. 3
  4. 4
  5. 7
Worked answer

3

Add the equations to eliminate y: 3x = 12, so x = 4.

Substitute back into x-y = 1: 4-y = 1, so y = 3.

Check in the first equation: 2(4)+3 = 11.

Problem 4

A system has the solution x=3 and y=5. How is that written as a point?

  1. (3,5)
  2. (5,3)
  3. (8,2)
  4. (3,3)
  5. (0,0)
Worked answer

(3,5)

A point is written with the x value first and the y value second.

So the solution is the point (3,5), where the two lines cross.

Problem 5

If 3x+2y=12 and x=2, what is y?

  1. 2
  2. 3
  3. 4
  4. 6
  5. 9
Worked answer

3

Substitute x=2: 3(2)+2y = 12, so 6+2y = 12.

Subtract 6: 2y = 6, so y = 3.

Problem 6

Two numbers add to 20 and differ by 6. What is the larger number?

  1. 7
  2. 10
  3. 13
  4. 14
  5. 26
Worked answer

13

Call them a and b with a+b = 20 and a-b = 6.

Adding the equations gives 2a = 26, so a = 13.

Then b = 7. Check: 13+7 = 20 and 13-7 = 6.

Problem 7

Where do the lines y=x+1 and y=2x-1 cross?

  1. (1,2)
  2. (2,3)
  3. (3,4)
  4. (0,1)
  5. (2,1)
Worked answer

(2,3)

At the crossing point both equations give the same y, so set them equal: x+1 = 2x-1.

Subtract x and add 1: x = 2.

Then y = 2+1 = 3, so the lines cross at (2,3).

Problem 8

Two lines in a system are parallel and distinct. How many solutions does the system have?

  1. None
  2. One
  3. Two
  4. Infinitely many
  5. It cannot be decided
Worked answer

None

A solution of the system is a point lying on both lines.

Distinct parallel lines never meet, so no point lies on both.

The system therefore has no solution. Infinitely many solutions would mean the two equations described the same line.

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