Solving Systems of Equations Worksheets with Two Variables - Free Printable

Let’s solve each system of equations by graphing. We’ll find where the two lines cross — that point (x, y) is the solution.

We’ll go one problem at a time.

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Problem 1:
Equations:
- y = -½x + 4
- 2x + 3y = -6

First, let’s rewrite the second equation in slope-intercept form (y = mx + b) so it’s easier to graph.

Start with:
2x + 3y = -6
Subtract 2x from both sides:
3y = -2x - 6
Divide by 3:
y = (-2/3)x - 2

Now we have:
Line 1: y = -½x + 4 → slope = -½, y-intercept = 4
Line 2: y = (-2/3)x - 2 → slope = -2/3, y-intercept = -2

Let’s pick some x-values and compute y for each line to plot points.

For Line 1 (y = -½x + 4):
- If x = 0 → y = 4 → point (0, 4)
- If x = 2 → y = -1 + 4 = 3 → point (2, 3)
- If x = 4 → y = -2 + 4 = 2 → point (4, 2)

For Line 2 (y = (-2/3)x - 2):
- If x = 0 → y = -2 → point (0, -2)
- If x = 3 → y = -2 - 2 = -4 → point (3, -4)
- If x = -3 → y = 2 - 2 = 0 → point (-3, 0)

Now, if we graph these, do they intersect? Let’s try solving algebraically to check.

Set the two expressions for y equal:

-½x + 4 = (-2/3)x - 2

Multiply everything by 6 to eliminate fractions:

6*(-½x) + 6*4 = 6*(-2/3x) + 6*(-2)
→ -3x + 24 = -4x - 12

Add 4x to both sides:
x + 24 = -12

Subtract 24:
x = -36

Wait — that seems too big. Let me double-check.

Original equation after setting equal:

-½x + 4 = (-2/3)x - 2

Move all terms to left side:

-½x + 4 + (2/3)x + 2 = 0
Combine like terms:

(-½ + 2/3)x + 6 = 0

Find common denominator for fractions: 6

-3/6 + 4/6 = 1/6

So: (1/6)x + 6 = 0
→ (1/6)x = -6
→ x = -36

Then plug into first equation: y = -½*(-36) + 4 = 18 + 4 = 22

So solution is (-36, 22)? That can’t be right on the given grid — grids only go to ±10 or so.

I think I made a mistake in rewriting the second equation.

Check original: 2x + 3y = -6

If x=0, 3y=-6 → y=-2 ✔️
If y=0, 2x=-6 → x=-3 ✔️

But when I solved algebraically, I got x=-36, which is way off the chart. That suggests maybe the lines don’t intersect within the visible grid? But that doesn’t make sense for this worksheet.

Wait — perhaps I misread the first equation? The image says “y = -½x + 4” — yes.

Maybe the second equation is written differently? Let me re-read.

Actually, looking back — the user wrote: “2x + 3y = -6” — but in the image, it might be different? Wait no — the user transcribed it as such.

Alternatively, maybe I should just graph using intercepts.

For Line 1: y = -½x + 4
x-intercept: set y=0 → 0 = -½x + 4 → ½x = 4 → x=8 → (8,0)
y-intercept: (0,4)

For Line 2: 2x + 3y = -6
x-intercept: y=0 → 2x=-6 → x=-3 → (-3,0)
y-intercept: x=0 → 3y=-6 → y=-2 → (0,-2)

Plotting these:

Line 1 goes from (0,4) to (8,0) — downward sloping gently.
Line 2 goes from (-3,0) to (0,-2) — steeper downward.

Do they meet? Let’s see — at x=6, Line 1: y = -3 + 4 = 1
Line 2: y = (-2/3)(6) - 2 = -4 -2 = -6 → not close.

At x= -6:
Line 1: y = -½*(-6)+4 = 3+4=7
Line 2: y = (-2/3)*(-6)-2 = 4 -2 = 2

Still not meeting.

Perhaps there's a typo in my assumption. Let me try solving again carefully.

Set:

y = -0.5x + 4
and
2x + 3y = -6

Substitute y from first into second:

2x + 3(-0.5x + 4) = -6
2x -1.5x + 12 = -6
0.5x + 12 = -6
0.5x = -18
x = -36

Same result. So mathematically, the solution is (-36, 22), but that’s not on the graph provided. This suggests either:

1. There’s a mistake in the problem transcription.
2. Or perhaps the second equation is different.

Looking back at the user’s input: “2x + 3y = -6” — but in many worksheets, sometimes it’s written as “2x - 3y = -6” or something else.

Wait — let me check Problem 1 again as written by user:

“1) y = -½x + 4; 2x + 3y = -6”

But if I assume it’s “2x - 3y = -6”, then:

2x - 3y = -6
-3y = -2x -6
y = (2/3)x + 2

Then set equal to y = -½x + 4:

-½x + 4 = (2/3)x + 2

Multiply by 6:

-3x + 24 = 4x + 12
24 - 12 = 4x + 3x
12 = 7x
x = 12/7 ≈ 1.71

Then y = -½*(12/7) + 4 = -6/7 + 28/7 = 22/7 ≈ 3.14

That would be around (1.7, 3.1) — which could fit on the grid.

But the user wrote “2x + 3y = -6”. Maybe it’s correct, and the intersection is outside the grid? But that seems unlikely for a basic graphing worksheet.

Alternatively, perhaps the first equation is y = -2x + 4? Let me test that.

If y = -2x + 4 and 2x + 3y = -6

Substitute:

2x + 3(-2x + 4) = -6
2x -6x + 12 = -6
-4x = -18
x = 4.5

y = -2(4.5) + 4 = -9 + 4 = -5

Point (4.5, -5) — still not great for small grid.

Another possibility: maybe the second equation is x + 3y = -6?

Let’s try that.

x + 3y = -6 → 3y = -x -6 → y = (-1/3)x -2

Set equal to y = -½x + 4:

-½x + 4 = (-1/3)x -2

Multiply by 6:

-3x + 24 = -2x -12
24 +12 = -2x +3x
36 = x

Again large number.

This is frustrating. Perhaps I should look at the other problems first and come back.

Let’s do Problem 2, which might be clearer.

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Problem 2:
Equations:
- y = ¾x - 3
- ⅓x - ¼y = -8

First, rewrite the second equation in slope-intercept form.

⅓x - ¼y = -8

Multiply every term by 12 to eliminate denominators:

12*(⅓x) - 12*(¼y) = 12*(-8)
4x - 3y = -96

Now solve for y:

-3y = -4x -96
y = (4/3)x + 32

That’s a very steep line with high y-intercept — again, likely outside the grid.

Set equal to first equation:

¾x - 3 = (4/3)x + 32

Multiply by 12:

9x - 36 = 16x + 384
-36 - 384 = 16x - 9x
-420 = 7x
x = -60

Even worse.

This suggests that either the equations are meant to be solved algebraically, or there’s a typo in the user’s transcription.

Perhaps the second equation in Problem 2 is ⅓x - ¼y = -1 or something smaller.

Let me try assuming it’s ⅓x - ¼y = -1

Then multiply by 12: 4x - 3y = -12
-3y = -4x -12
y = (4/3)x + 4

Set equal to y = ¾x - 3:

¾x - 3 = (4/3)x + 4

Multiply by 12:

9x - 36 = 16x + 48
-36 -48 = 16x -9x
-84 = 7x
x = -12

Still large.

Try if it’s ⅓x - ¼y = 0

Then 4x - 3y = 0 → y = (4/3)x

Set to ¾x - 3 = (4/3)x

9x - 36 = 16x → -36 = 7x → x≈-5.14

Better, but still not nice numbers.

Perhaps the first equation is y = -¾x -3? Let's stop guessing.

Another idea: maybe the "graph" part means to estimate from the grid, and the equations are designed to intersect at integer points within the grid.

Let me try Problem 3.

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Problem 3:
Equations:
- y = x + 7
- y = -½x - 3½ (which is -3.5)

Set equal:

x + 7 = -½x - 3.5

Add ½x to both sides:

1.5x + 7 = -3.5

Subtract 7:

1.5x = -10.5

x = -10.5 / 1.5 = -7

Then y = -7 + 7 = 0

So solution is (-7, 0)

That’s a nice integer point! And likely on the grid.

Let me verify with second equation: y = -½*(-7) - 3.5 = 3.5 - 3.5 = 0 ✔️

Perfect.

So for Problem 3, solution is (-7, 0)

Now, why did Problems 1 and 2 give large numbers? Probably because of transcription errors.

Looking back at user’s input for Problem 1: “y = -½x + 4; 2x + 3y = -6”

But if it were “y = -2x + 4” and “2x + 3y = 6” or something.

Perhaps in the actual image, the equations are different.

Since the user said “upload an image”, but I can't see it, I have to rely on their text.

But for Problem 3, it worked perfectly.

Let me try Problem 4.

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Problem 4:
Equations:
- y = -⅔x + 2
- y = ½x + ½

Set equal:

-⅔x + 2 = ½x + ½

Get all x terms on one side.

Add ⅔x to both sides:

2 = (½ + ⅔)x + ½

Compute ½ + ⅔ = 3/6 + 4/6 = 7/6

So:

2 = (7/6)x + ½

Subtract ½:

1.5 = (7/6)x

Or 3/2 = (7/6)x

Multiply both sides by 6/7:

x = (3/2)*(6/7) = 18/14 = 9/7 ≈ 1.2857

Then y = ½*(9/7) + ½ = 9/14 + 7/14 = 16/14 = 8/7 ≈ 1.1429

Not integer, but possible.

Check with first equation: y = -⅔*(9/7) + 2 = -18/21 + 2 = -6/7 + 14/7 = 8/7 ✔️

So solution is (9/7, 8/7) or approximately (1.29, 1.14)

On a grid, you'd estimate.

But let's see if we can write as fraction.

Final Answer for Problem 4: (9/7, 8/7)

Now Problem 5.

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Problem 5:
Equations:
- y = -⅓x - 14
- x - 3y = -18

Rewrite second equation:

x - 3y = -18
-3y = -x -18
y = (1/3)x + 6

Set equal to first:

-⅓x - 14 = (1/3)x + 6

Add ⅓x to both sides:

-14 = (2/3)x + 6

Subtract 6:

-20 = (2/3)x

Multiply both sides by 3/2:

x = -20 * 3/2 = -30

Then y = (1/3)*(-30) + 6 = -10 + 6 = -4

Solution: (-30, -4) — again large.

But if the first equation was y = -⅓x + 14 or something.

Assume it's y = -⅓x + 14

Then set to y = (1/3)x + 6

-⅓x + 14 = (1/3)x + 6

14 - 6 = (1/3)x + ⅓x

8 = (2/3)x

x = 8 * 3/2 = 12

y = (1/3)(12) + 6 = 4 + 6 = 10

Point (12,10) — still large for typical grid.

If first equation is y = -3x -14, etc.

Perhaps for Problem 5, it's y = -⅓x - 4 or something.

Let's calculate with given:

y = -⅓x - 14
y = (1/3)x + 6

Set equal:

-⅓x - 14 = (1/3)x + 6

As above, x = -30, y = -4

Not good.

Another possibility: the second equation is x + 3y = -18

Then 3y = -x -18, y = (-1/3)x -6

Set to y = -⅓x -14

-⅓x -14 = -⅓x -6

Then -14 = -6, contradiction — parallel lines, no solution.

Not likely.

Perhaps it's x - 3y = 18

Then -3y = -x +18, y = (1/3)x -6

Set to y = -⅓x -14

-⅓x -14 = (1/3)x -6

-14 +6 = (1/3)x + ⅓x

-8 = (2/3)x

x = -12

y = (1/3)(-12) -6 = -4 -6 = -10

Still large.

I think there might be typos in the user's transcription for some problems.

Let's do Problem 6.

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Problem 6:
Equations:
- y = 2x - 4
- y = -x + 5

Set equal:

2x - 4 = -x + 5

Add x to both sides:

3x - 4 = 5

Add 4:

3x = 9

x = 3

y = 2*3 - 4 = 6 - 4 = 2

Or y = -3 + 5 = 2

Solution: (3, 2) — perfect, integer, on grid.

Great.

So summarizing what we have:

Problem 3: (-7, 0)
Problem 6: (3, 2)
Problem 4: (9/7, 8/7) — but let's keep as fraction.

For Problem 1,2,5, we have issues.

Perhaps in the actual image, the equations are:

For Problem 1: maybe y = -2x + 4 and 2x + y = 4 or something.

Since the user might have mistyped, and for the sake of completing, I'll assume that for Problems 1,2,5, the intended solutions are within grid, and perhaps I can infer.

Another approach: for graphing, sometimes they want you to plot and estimate, but since this is text, I need exact.

Let's look at Problem 2 again.

User wrote: "y = ¾x - 3; ⅓x - ¼y = -8"

But if it's "⅓x - ¼y = -1" , we had x= -12, still bad.

If "⅓x - ¼y = 0", x≈-5.14

If "⅓x - ¼y = 1", then 4x -3y = 12, y = (4/3)x -4

Set to ¾x -3 = (4/3)x -4

9x -36 = 16x -48

-36 +48 = 16x -9x

12 = 7x

x=12/7≈1.71, y= ¾*(12/7) -3 = 9/7 - 21/7 = -12/7

Point (12/7, -12/7) — symmetric.

Possible.

But let's notice that in Problem 4, we have nice fractions.

Perhaps for consistency, I'll provide the mathematical solutions based on given equations, even if large.

But for a student, that might be confusing.

Maybe the "graph" is to be done on paper, and the answer is to be found by plotting, so for those with large solutions, they might not intersect on the grid, but that seems odd.

Another idea: perhaps the second equation in Problem 1 is "2x - 3y = -6" as I thought earlier.

Let me assume that for Problem 1: y = -½x + 4 and 2x - 3y = -6

Then from 2x - 3y = -6, -3y = -2x -6, y = (2/3)x + 2

Set equal:

-½x + 4 = (2/3)x + 2

Multiply by 6:

-3x + 24 = 4x + 12

24 - 12 = 4x + 3x

12 = 7x

x = 12/7

y = -½*(12/7) + 4 = -6/7 + 28/7 = 22/7

So (12/7, 22/7) ≈ (1.71, 3.14)

Similarly, for Problem 2, if we assume the constant is different.

But to move forward, I'll use the calculations as per given, and for the final answer, list them.

Perhaps the user can clarify, but since I must answer, I'll proceed with the math.

Let's list all:

Problem 1:
y = -0.5x + 4
2x + 3y = -6 → y = (-2/3)x -2
Set equal: -0.5x + 4 = -2/3 x -2
As before, x = -36, y = 22
Solution: (-36, 22)

Problem 2:
y = 0.75x - 3
⅓x - ¼y = -8 → as above, y = (4/3)x + 32
Set equal: 0.75x - 3 = 1.333x + 32
-3 -32 = 1.333x - 0.75x
-35 = 0.5833x
x = -35 / (7/12) wait earlier calculation gave x= -60
From before: x = -60, y = 0.75*(-60) -3 = -45 -3 = -48
Solution: (-60, -48)

Problem 3: (-7, 0) — good

Problem 4: (9/7, 8/7)

Problem 5:
y = -1/3 x -14
x - 3y = -18 → y = (1/3)x +6
Set equal: -1/3 x -14 = 1/3 x +6
-14 -6 = 1/3 x +1/3 x
-20 = 2/3 x
x = -30
y = 1/3*(-30) +6 = -10+6= -4
Solution: (-30, -4)

Problem 6: (3,2) — good

This is not satisfactory for a graphing worksheet.

Perhaps for Problem 5, the first equation is y = -1/3 x + 4 or something.

Let's assume that in the actual image, the equations are designed to have nice solutions.

For example, common systems:

Like for Problem 1, perhaps it's y = -2x + 4 and y = x -2 or something.

I recall that in many worksheets, they use:

For instance, y = x + 2 and y = -x + 4 intersect at (1,3)

But here, let's look at Problem 3 and 6 which worked.

For Problem 4, (9/7, 8/7) is fine.

For Problem 2, if we take y = 3/4 x - 3 and say 1/3 x - 1/4 y = 0, then y = 4/3 x, set 3/4 x -3 = 4/3 x, etc, not good.

Another guess: perhaps "⅓x - ¼y = -1" and y = 3/4 x - 3

Then from ⅓x - ¼y = -1, multiply by 12: 4x -3y = -12, y = (4/3)x +4

Set to 3/4 x -3 = 4/3 x +4

9x -36 = 16x +48

-36-48 = 16x-9x

-84 = 7x

x= -12, y= 3/4*(-12) -3 = -9-3= -12

Point (-12,-12) — still large.

I think I have to accept the mathematical solutions.

Perhaps the "graph" is to be done, and for those with large solutions, they are parallel or something, but in this case, they do intersect, just far away.

For the sake of the student, I'll provide the solutions as calculated, and suggest that for graphing, they may need larger grid or check the equations.

But let's double-check Problem 1 with a different approach.

Suppose we graph y = -0.5x + 4 and 2x + 3y = -6.

Let me find two points for each.

Line 1: when x=0, y=4; x=2, y=3; x=4, y=2; x=6, y=1; x=8, y=0; x=10, y= -1

Line 2: 2x + 3y = -6

When x=0, 3y= -6, y= -2

When y=0, 2x= -6, x= -3

When x=3, 6 +3y= -6, 3y= -12, y= -4

When x= -6, -12 +3y= -6, 3y=6, y=2

So points: (0,-2), (-3,0), (3,-4), (-6,2)

Now, does line 1 and line 2 intersect? From the points, line 1 is decreasing slowly, line 2 is decreasing faster.

At x= -6, line 1: y = -0.5*(-6) +4 = 3+4=7

Line 2: y=2 (from above)

At x=0, line 1: y=4, line 2: y= -2

At x=6, line 1: y=1, line 2: y= -4 -2? Earlier for x=6, 2*6 +3y= -6, 12+3y= -6, 3y= -18, y= -6

So at x=6, line 1: y=1, line 2: y= -6

The difference is increasing, so they intersect when x is negative large.

For example, at x= -12, line 1: y = -0.5*(-12) +4 = 6+4=10

Line 2: 2*(-12) +3y = -6, -24 +3y = -6, 3y=18, y=6

At x= -24, line 1: y = 12 +4=16

Line 2: 2*(-24) +3y = -6, -48 +3y = -6, 3y=42, y=14

At x= -36, line 1: y = 18 +4=22

Line 2: 2*(-36) +3y = -6, -72 +3y = -6, 3y=66, y=22

Yes, so at x= -36, y=22 for both.

So mathematically correct.

For a graphing exercise, this is impractical, so likely a typo in the problem.

Given that, and since the user might have copied wrong, but for accuracy, I'll use the calculations.

Perhaps in the image, it's "2x - 3y = 6" or something.

To resolve, I'll provide the solutions as per given equations, and for the final answer, list them.

So:

Problem 1: (-36, 22)

Problem 2: (-60, -48)

Problem 3: (-7, 0)

Problem 4: (9/7, 8/7)

Problem 5: (-30, -4)

Problem 6: (3, 2)

But for Problem 4, 9/7 and 8/7 can be left as fractions.

Now for the response, since the user asked to solve, I'll box the answers.

But let's confirm Problem 4 again.

y = -2/3 x + 2

y = 1/2 x + 1/2

Set equal:

-2/3 x + 2 = 1/2 x + 1/2

Bring all to left:

-2/3 x - 1/2 x + 2 - 1/2 = 0

Common denominator for x terms: 6

-4/6 x - 3/6 x + 3/2 = 0

-7/6 x + 3/2 = 0

7/6 x = 3/2

x = (3/2) * (6/7) = 18/14 = 9/7

y = 1/2 * 9/7 + 1/2 = 9/14 + 7/14 = 16/14 = 8/7

Yes.

So final answers:

1) (-36, 22)

2) (-60, -48)

3) (-7, 0)

4) (9/7, 8/7)

5) (-30, -4)

6) (3, 2)

I think that's it.

For the student, I can say that for some problems, the intersection is outside the typical grid, so they might need to adjust or check the equations.

But for now, I'll provide these.

Final Answer:

1) \boxed{(-36,\ 22)}
2) \boxed{(-60,\ -48)}
3) \boxed{(-7,\ 0)}
4) \boxed{\left(\frac{9}{7},\ \frac{8}{7}\right)}
5) \boxed{(-30,\ -4)}
6) \boxed{(3,\ 2)}