Solving Systems of Linear Equations using the Substitution Method worksheet.

Let’s solve each system of linear equations using the substitution method.

We’ll go one by one. The substitution method means:
1. Solve one equation for one variable (like x or y).
2. Plug that expression into the other equation.
3. Solve for the remaining variable.
4. Plug back to find the first variable.

---

Problem 1:
Equations:
① x + y = 6
② y = 2x + 3

Step 1: Equation ② already gives us y in terms of x → y = 2x + 3
Step 2: Substitute into ①:
x + (2x + 3) = 6
→ 3x + 3 = 6
→ 3x = 3
→ x = 1

Step 3: Plug x = 1 into ②:
y = 2(1) + 3 = 5

✔ Solution: (1, 5) → Matches answer key #1: (1, 5)

---

Problem 2:
Equations:
① 3x - y = 7
② y = 2x - 4

Step 1: Use ② directly: y = 2x - 4
Step 2: Plug into ①:
3x - (2x - 4) = 7
→ 3x - 2x + 4 = 7
→ x + 4 = 7
→ x = 3

Step 3: Plug x = 3 into ②:
y = 2(3) - 4 = 6 - 4 = 2

✔ Solution: (3, 2) → Matches answer key #2: (3, 2)

Wait — answer key says “(-1, -6)” for problem 2? That doesn’t match our work. Let me double-check.

Hold on — maybe I misread the problem numbers? Let’s check the image again mentally.

Actually, looking at the original list:

The problems are numbered 1 through 8, and answers are labeled A through H.

Answer key says:
A. (6, -1)
B. (-1, -6) ← this is listed as answer for problem 2? But we got (3,2) for problem 2.

Wait — perhaps the problem numbering in the worksheet does NOT match the answer lettering order? Or maybe I misassigned which problem corresponds to which answer.

Looking back at user input:

It says:

“Answers” column has:

1. A. (6, -1)
2. B. (-1, -6)
3. C. (1, 5)
4. D. (5, -2)
5. E. (5, -1)
6. F. (0, 4)
7. G. (1, -9)
8. H. (-1, -2)

But in the left column, problems are numbered 1 to 8.

So likely:

Problem 1 → Answer A? But we solved Problem 1 and got (1,5), which is answer C.

Ah! There's a mismatch. Probably the "Answers" column is not aligned with problem number — it’s just a list of possible answers, and we have to match them.

Actually, re-reading the top: “Solve each system... Answers” then lists 1. A. (6,-1) etc. — that might mean:

For problem 1, the correct answer is among A-H, and they’re listing what the answer should be for each problem? But that would mean:

Problem 1 → Answer is A: (6,-1)? But we calculated (1,5).

That can't be right unless I made a mistake.

Wait — let me re-solve Problem 1 carefully.

Problem 1:
x + y = 6
y = 2x + 3

Substitute: x + (2x+3) = 6 → 3x + 3 = 6 → 3x=3 → x=1 → y=2(1)+3=5 → (1,5)

But according to the answer key shown, problem 1 is supposed to be A: (6,-1)? That doesn’t make sense.

Unless... perhaps the “Answers” column is mislabeled? Or maybe the problems are not in order?

Wait — look at problem 3:

Problem 3:
3x - 5y = 12
y = 2x - 4

Let’s solve this.

From ②: y = 2x - 4
Plug into ①: 3x - 5(2x - 4) = 12
→ 3x -10x + 20 = 12
→ -7x + 20 = 12
→ -7x = -8
→ x = 8/7 → not integer. But all answers are integers. So probably not.

Wait — maybe I need to match the problems to the correct answers based on calculation, ignoring the “1. A.” labeling? Because that seems off.

Alternatively, perhaps the “Answers” column is meant to be matched after solving — meaning, for problem 1, you get an answer, and you pick which letter it matches from A-H.

But the way it’s written: “1. A. (6,-1)” suggests that for problem 1, the answer is A.

But our calculation for problem 1 gives (1,5), which is option C.

This is confusing.

Perhaps there’s a typo in how the answers are presented? Or maybe I should ignore the “1. A.” part and just solve each problem and report the solution pair.

Since the instruction is to “solve the problem accurately”, and the final answer should be the correct solution, I will solve each problem correctly and provide the ordered pair.

Then, if needed, match to the letters, but since the question doesn’t ask to match letters, just to solve, I’ll provide the solutions.

But the user included the answer key, so probably expects us to verify or use it.

Wait — let’s try Problem 2 again as written:

Problem 2:
3x - y = 7
y = 2x - 4

As before: substitute → 3x - (2x - 4) = 7 → x + 4 = 7 → x=3, y=2 → (3,2)

But (3,2) is not in the answer list? Answer list has:

A. (6,-1)
B. (-1,-6)
C. (1,5)
D. (5,-2)
E. (5,-1)
F. (0,4)
G. (1,-9)
H. (-1,-2)

(3,2) is not there. That means either I misread the problem, or the problem is different.

Let me look back at the user’s text representation:

In the initial prompt, under “Problems”:

1.
x + y = 6
y = 2x + 3

2.
3x - y = 7
y = 2x - 4

3.
3x - 5y = 12
y = 2x - 4

4.
2x + 4y = 8
y = 2x + 6

5.
2x + 3y = 4
y = 2x - 6

6.
2x + 3y = 16
3x + 2y = 26

7.
5x - 2y = 11
3x + y = -1

8.
4x - 3y = -1
2x + y = -5

And answers:

1. A. (6, -1)
2. B. (-1, -6)
3. C. (1, 5)
4. D. (5, -2)
5. E. (5, -1)
6. F. (0, 4)
7. G. (1, -9)
8. H. (-1, -2)

Now, let’s solve Problem 3:

Problem 3:
3x - 5y = 12
y = 2x - 4

Substitute y into first equation:

3x - 5(2x - 4) = 12
3x -10x + 20 = 12
-7x + 20 = 12
-7x = -8
x = 8/7 → not integer. But all answers are integers. So likely, this is not matching.

Perhaps the second equation is different? Or maybe it's y = 2x + 4? Let's assume it's as written.

Maybe for Problem 3, it's supposed to be something else.

Another idea: perhaps the "Answers" column is not for the problems in order, but rather a list of possible answers, and we need to solve and see which one matches.

But the way it's formatted, it looks like for problem 1, answer is A, etc.

Let’s try Problem 4:

Problem 4:
2x + 4y = 8
y = 2x + 6

Substitute y into first equation:

2x + 4(2x + 6) = 8
2x + 8x + 24 = 8
10x + 24 = 8
10x = -16
x = -1.6 → not integer.

Not matching any answer.

This is strange. Perhaps there's a typo in the problem transcription.

Let me try Problem 5:

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

Substitute:

2x + 3(2x - 6) = 4
2x + 6x - 18 = 4
8x - 18 = 4
8x = 22
x = 22/8 = 11/4 → not integer.

Still not working.

Problem 6:
2x + 3y = 16
3x + 2y = 26

This one doesn't have y isolated, so we need to solve one for a variable.

Let’s solve the first equation for x:

2x = 16 - 3y
x = (16 - 3y)/2

Plug into second equation:

3*(16 - 3y)/2 + 2y = 26
Multiply both sides by 2 to eliminate denominator:

3(16 - 3y) + 4y = 52
48 - 9y + 4y = 52
48 - 5y = 52
-5y = 4
y = -4/5 → not integer.

Not matching.

Problem 7:
5x - 2y = 11
3x + y = -1

Solve second equation for y: y = -1 - 3x

Plug into first:

5x - 2(-1 - 3x) = 11
5x +2 +6x = 11
11x +2 = 11
11x = 9
x = 9/11 → not integer.

Problem 8:
4x - 3y = -1
2x + y = -5

Solve second for y: y = -5 - 2x

Plug into first:

4x - 3(-5 - 2x) = -1
4x +15 +6x = -1
10x +15 = -1
10x = -16
x = -1.6 → not integer.

None of these are giving integer solutions except Problems 1 and 2, which gave (1,5) and (3,2), but (3,2) is not in the answer list.

This suggests that either the problems are transcribed incorrectly, or the answer key is for different problems.

Perhaps the "Answers" column is misaligned. Let's calculate all solutions and see which ones match the given answer pairs.

From earlier:

Problem 1: (1,5) → matches C

Problem 2: (3,2) → not in list

But let's try Problem 2 with a different interpretation. What if the second equation is y = 2x + 4? Then:

3x - (2x +4) = 7 → x -4 =7 → x=11, y=26 — no.

Or if it's y = -2x -4? Then 3x - (-2x -4) =7 → 3x +2x +4=7 → 5x=3 → x=0.6 — no.

Perhaps for Problem 2, it's 3x + y =7 and y=2x-4? Then 3x + (2x-4)=7 → 5x=11 — no.

Another idea: perhaps the answer key is correct, and I need to solve to match those answers.

Let's take answer B: (-1, -6) and see which problem it satisfies.

Try Problem 2: 3x - y =7, y=2x-4

If x=-1, y=2*(-1)-4= -2-4= -6

Then 3*(-1) - (-6) = -3 +6 =3 ≠7 — not satisfied.

Try Problem 3: 3x -5y=12, y=2x-4

x=-1, y=2*(-1)-4= -6

3*(-1