Solving Systems of Equations - Graphing Worksheet

To solve these systems of equations by graphing, we need to find the point where the two lines intersect. That point (x, y) is the solution because it satisfies both equations.

Let’s go step by step for each problem:

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Problem 1:
Equations:
- y = x + 2
- y = -2x + 2

Step 1: Graph both lines.
- First line: slope = 1, y-intercept = 2 → starts at (0,2), go up 1, right 1.
- Second line: slope = -2, y-intercept = 2 → starts at (0,2), go down 2, right 1.

They both start at (0,2). Let’s check if they intersect elsewhere.

Set them equal:
x + 2 = -2x + 2
→ 3x = 0 → x = 0
Then y = 0 + 2 = 2

So solution is (0, 2)

✔ Verified.

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Problem 2:
Equations:
- y = (2/3)x
- y = -(4/3)x + 6

Set equal:
(2/3)x = -(4/3)x + 6
Multiply all terms by 3:
2x = -4x + 18
6x = 18 → x = 3
Then y = (2/3)(3) = 2

Solution: (3, 2)

✔ Verified.

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Problem 3:
Equations:
- y = (1/5)x + 1
- y = -(2/5)x - 4

Set equal:
(1/5)x + 1 = -(2/5)x - 4
Multiply by 5:
x + 5 = -2x - 20
3x = -25 → x = -25/3 ≈ -8.333...

Wait — let’s double-check:

(1/5)x + 1 = -(2/5)x - 4
Add (2/5)x to both sides:
(3/5)x + 1 = -4
Subtract 1:
(3/5)x = -5
Multiply by 5/3:
x = -5 * (5/3) = -25/3 → yes.

Then y = (1/5)(-25/3) + 1 = -5/3 + 1 = -2/3

Solution: (-25/3, -2/3)

But since this is a graphing worksheet, maybe they expect integer solutions? Let me check if I copied correctly.

Original:
y = (1/5)x + 1
y = -(2/5)x - 4

Yes. So fractional answer is correct.

But let’s verify with second equation:
y = -(2/5)(-25/3) - 4 = (50/15) - 4 = 10/3 - 12/3 = -2/3 → matches.

So solution is (-25/3, -2/3)

However, looking at the grid, it might be hard to plot exactly — but mathematically, that’s correct.

Wait — perhaps I made a sign error?

Let me re-solve:

(1/5)x + 1 = -(2/5)x - 4
Bring like terms together:
(1/5)x + (2/5)x = -4 - 1
(3/5)x = -5
x = -5 * (5/3) = -25/3 → same.

Okay, so it’s correct.

But let’s hold on — maybe the problem expects us to graph and estimate? But since we’re solving accurately, we’ll keep the exact value.

Actually, let me check Problem 4 first — maybe there’s a pattern.

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

Set equal:
-2x + 2 = -4x - 2
Add 4x:
2x + 2 = -2
2x = -4 → x = -2
Then y = -2(-2) + 2 = 4 + 2 = 6

Check with second: y = -4(-2) - 2 = 8 - 2 = 6 → good.

Solution: (-2, 6)

✔ Verified.

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Problem 5:
Equations:
- y = (2/7)x - 4
- y = -x + 5

Set equal:
(2/7)x - 4 = -x + 5
Add x:
(2/7)x + x - 4 = 5 → (9/7)x - 4 = 5
Add 4:
(9/7)x = 9
Multiply by 7/9:
x = 9 * (7/9) = 7

Then y = -7 + 5 = -2

Check with first: y = (2/7)(7) - 4 = 2 - 4 = -2 → good.

Solution: (7, -2)

✔ Verified.

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Problem 6:
Equations:
- 5x - 4y = 16
- -3x + 2y = -12

We can solve by substitution or elimination. Let’s use elimination.

Multiply second equation by 2:
-6x + 4y = -24

Now add to first equation:
5x - 4y = 16
+ (-6x + 4y = -24)
→ -x = -8 → x = 8

Plug into second original:
-3(8) + 2y = -12 → -24 + 2y = -12 → 2y = 12 → y = 6

Check first: 5(8) - 4(6) = 40 - 24 = 16 → good.

Solution: (8, 6)

✔ Verified.

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Problem 7:
Equations:
- y = (1/3)x - 1
- 5x + 3y = 15

Substitute first into second:

5x + 3[(1/3)x - 1] = 15
5x + x - 3 = 15
6x = 18 → x = 3

Then y = (1/3)(3) - 1 = 1 - 1 = 0

Check second: 5(3) + 3(0) = 15 → good.

Solution: (3, 0)

✔ Verified.

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Problem 8:
Equations:
- -x + 2y = 6
- -3x + 2y = 14

Subtract first from second:

(-3x + 2y) - (-x + 2y) = 14 - 6
-3x + 2y + x - 2y = 8
-2x = 8 → x = -4

Plug into first: -(-4) + 2y = 6 → 4 + 2y = 6 → 2y = 2 → y = 1

Check second: -3(-4) + 2(1) = 12 + 2 = 14 → good.

Solution: (-4, 1)

✔ Verified.

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Now back to Problem 3 — we had:

y = (1/5)x + 1
y = -(2/5)x - 4

Solution: x = -25/3, y = -2/3

But let me write it as mixed number: -8 1/3 and -2/3 — still messy.

Perhaps I should leave it as fractions.

Alternatively, maybe the problem has a typo? But based on given, it’s correct.

Wait — let me try plugging in x = -5:

First equation: y = (1/5)(-5) + 1 = -1 + 1 = 0
Second: y = -(2/5)(-5) - 4 = 2 - 4 = -2 → not same.

x = -10:
First: y = (1/5)(-10)+1 = -2+1 = -1
Second: y = -(2/5)(-10)-4 = 4 - 4 = 0 → no.

x = -25/3 ≈ -8.333 — let’s compute numerically:

x = -8.333
y1 = (1/5)(-8.333) + 1 ≈ -1.666 + 1 = -0.666
y2 = -(2/5)(-8.333) - 4 ≈ (3.333) - 4 = -0.667 → close enough, rounding error.

So yes, correct.

But for the purpose of this worksheet, perhaps they expect us to graph and see intersection — but since we’re solving accurately, we’ll report exact values.

However, looking back at the user’s request: “Solve the problem accurately.” And “verify your answer”.

So for Problem 3, solution is (-25/3, -2/3)

But let me write all solutions clearly.

Final Answers:

1. (0, 2)
2. (3, 2)
3. (-25/3, -2/3)
4. (-2, 6)
5. (7, -2)
6. (8, 6)
7. (3, 0)
8. (-4, 1)

All verified.

Now, since the instruction says “ONLY THEN provide the final answer”, and we’ve done all reasoning, here’s the final answer section.

Final Answer:
1. (0, 2)
2. (3, 2)
3. (-25/3, -2/3)
4. (-2, 6)
5. (7, -2)
6. (8, 6)
7. (3, 0)
8. (-4, 1)