Math worksheet matching systems of linear equations to their solutions.

Let’s solve each system of linear equations one by one to find the correct ordered pair (x, y) that satisfies both equations in the system.

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System 1:
2x + y = 7
3x - y = 3

Add the two equations to eliminate y:
(2x + y) + (3x - y) = 7 + 3 → 5x = 10 → x = 2
Plug x = 2 into first equation: 2(2) + y = 7 → 4 + y = 7 → y = 3
✔ Solution: (2, 3)

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System 2:
x - y = 4
x + y = 2

Add the two equations:
(x - y) + (x + y) = 4 + 2 → 2x = 6 → x = 3
Plug x = 3 into second equation: 3 + y = 2 → y = -1
✔ Solution: (3, -1)

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System 3:
2x + 3y = 8
3x + y = 5

Solve second equation for y: y = 5 - 3x
Substitute into first equation:
2x + 3(5 - 3x) = 8 → 2x + 15 - 9x = 8 → -7x = -7 → x = 1
Then y = 5 - 3(1) = 2
✔ Solution: (1, 2)

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System 4:
2x - 3y = 8
2x + 3y = 4

Add the two equations:
(2x - 3y) + (2x + 3y) = 8 + 4 → 4x = 12 → x = 3
Plug x = 3 into second equation: 2(3) + 3y = 4 → 6 + 3y = 4 → 3y = -2 → y = -2/3
✔ Solution: (3, -2/3)

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System 5:
5x + y = 15
3x + 2y = 9

Solve first equation for y: y = 15 - 5x
Substitute into second:
3x + 2(15 - 5x) = 9 → 3x + 30 - 10x = 9 → -7x = -21 → x = 3
Then y = 15 - 5(3) = 0
✔ Solution: (3, 0)

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System 6:
-x + 2y = 10
x - 3y = -13

Add the two equations:
(-x + 2y) + (x - 3y) = 10 + (-13) → -y = -3 → y = 3
Plug y = 3 into second equation: x - 3(3) = -13 → x - 9 = -13 → x = -4
✔ Solution: (-4, 3)

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System 7:
2x - 3y = 0
-2x + 2y = -2

Add the two equations:
(2x - 3y) + (-2x + 2y) = 0 + (-2) → -y = -2 → y = 2
Plug y = 2 into first equation: 2x - 3(2) = 0 → 2x = 6 → x = 3
✔ Solution: (3, 2)

Wait — let me double-check this. If x=3, y=2:
First equation: 2(3) - 3(2) = 6 - 6 = 0 ✔
Second: -2(3) + 2(2) = -6 + 4 = -2 ✔
Yes, correct. But look at the answer choices — we already have (3,2) from System 7? Wait no — actually, let’s check System 8 too.

Actually, wait — I think I made a mistake here. Let me recheck System 7 and 8 carefully.

Hold on — System 7 is:

2x - 3y = 0
-2x + 2y = -2

We got (3,2). But let’s see if that matches any other system? No, it’s unique.

But now System 8:

System 8:
y = 3x + 6
2y = -3x + 3

Substitute first into second:
2(3x + 6) = -3x + 3 → 6x + 12 = -3x + 3 → 9x = -9 → x = -1
Then y = 3(-1) + 6 = 3
✔ Solution: (-1, 3)

So System 7 gives (3,2), System 8 gives (-1,3)

Now let’s list all solutions again clearly:

1. (2,3)
2. (3,-1)
3. (1,2)
4. (3, -2/3)
5. (3,0)
6. (-4,3)
7. (3,2) ← Wait, but earlier I thought System 7 was (3,2), but let me verify with substitution again.

Wait — hold on! In System 7:

Equation 1: 2x - 3y = 0
Equation 2: -2x + 2y = -2

If I add them: (2x - 2x) + (-3y + 2y) = 0 - 2 → -y = -2 → y = 2
Then plug into Eq1: 2x - 3(2) = 0 → 2x = 6 → x = 3 → (3,2) ✔

But now looking at the answer boxes, there is a green box labeled (2,3) — which we assigned to System 1.

And purple box (3,2) — which should go to System 7.

But wait — let me check if I mixed up Systems 1 and 7?

No — System 1: 2x+y=7, 3x-y=3 → solved to (2,3) ✔
System 7: 2x-3y=0, -2x+2y=-2 → solved to (3,2) ✔

All good.

Now let’s match each system to its solution:

Top left: System 1 → (2,3) → green box
Top right: System 2 → (3,-1) → orange box
Middle left: System 3 → (1,2) → light blue box
Middle right: System 4 → (3, -2/3) → dark blue box
Bottom left top: System 5 → (3,0) → red box
Bottom left bottom: System 6 → (-4,3) → pink box
Bottom right top: System 7 → (3,2) → purple box
Bottom right bottom: System 8 → (-1,3) → red-orange box? Wait, there are two red boxes?

Looking back at the image description:

Boxes:

Red: (3,0)
Pink: (-4,3)
Red: (-1,3) ← another red? Or maybe different shade?
Purple: (3,2)
Orange: (3,-1)
Blue: (3,-2/3)
Green: (2,3)
Light blue: (1,2)

So:

System 1 → (2,3) → Green
System 2 → (3,-1) → Orange
System 3 → (1,2) → Light Blue
System 4 → (3, -2/3) → Dark Blue
System 5 → (3,0) → Red (first red)
System 6 → (-4,3) → Pink
System 7 → (3,2) → Purple
System 8 → (-1,3) → Second Red (or maybe it's labeled differently — but in text it says “(-1,3)” in a red box)

In the user’s image description, it lists:

[ (3,0) ] — red
[ (-4,3) ] — pink
[ (-1,3) ] — red (another red?)
[ (3,2) ] — purple
[ (3,-1) ] — orange
[ (3,-2/3) ] — blue
[ (2,3) ] — green
[ (1,2) ] — light blue

So yes, two red boxes: one for (3,0), one for (-1,3)

Final matching:

- Top-left system (2x+y=7, 3x−y=3) → (2,3) → Green
- Top-right system (x−y=4, x+y=2) → (3,−1) → Orange
- Middle-left system (2x+3y=8, 3x+y=5) → (1,2) → Light Blue
- Middle-right system (2x−3y=8, 2x+3y=4) → (3, −2/3) → Blue
- Bottom-left-top system (5x+y=15, 3x+2y=9) → (3,0) → Red (first)
- Bottom-left-bottom system (−x+2y=10, x−3y=−13) → (−4,3) → Pink
- Bottom-right-top system (2x−3y=0, −2x+2y=−2) → (3,2) → Purple
- Bottom-right-bottom system (y=3x+6, 2y=−3x+3) → (−1,3) → Red (second)

All verified.

Final Answer:
Top-left → (2,3)
Top-right → (3,-1)
Middle-left → (1,2)
Middle-right → (3,-2/3)
Bottom-left-top → (3,0)
Bottom-left-bottom → (-4,3)
Bottom-right-top → (3,2)
Bottom-right-bottom → (-1,3)