Solving systems of equations by elimination or by substitution ... - Free Printable

Let’s solve each system of equations using substitution. We’ll go one at a time, carefully.

---

Problem 1:
```
m + n = 3
m - n = 5
```

Step 1: Solve the first equation for m.
→ m = 3 - n

Step 2: Substitute into the second equation.
→ (3 - n) - n = 5
→ 3 - 2n = 5
→ -2n = 2
→ n = -1

Step 3: Plug back to find m.
→ m = 3 - (-1) = 4

✔ Check:
4 + (-1) = 3 ✔️
4 - (-1) = 5 ✔️

---

Problem 2:
```
2x + 3y = 32
3x + y = 10
```

Step 1: Solve the second equation for y.
→ y = 10 - 3x

Step 2: Substitute into first equation.
→ 2x + 3(10 - 3x) = 32
→ 2x + 30 - 9x = 32
→ -7x = 2
→ x = -2/7

Wait — that gives a fraction. Let’s double-check.

Actually, let’s try solving for y again:
From 3x + y = 10 → y = 10 - 3x ✔️

Plug in:
2x + 3(10 - 3x) = 32
2x + 30 - 9x = 32
-7x = 2 → x = -2/7

Then y = 10 - 3(-2/7) = 10 + 6/7 = 76/7

But this seems messy. Maybe I made an error? Let’s check with original equations:

2*(-2/7) + 3*(76/7) = -4/7 + 228/7 = 224/7 = 32 ✔️
3*(-2/7) + 76/7 = -6/7 + 76/7 = 70/7 = 10 ✔️

So it’s correct, even if fractions.

But wait — maybe we can avoid fractions by solving differently?

Alternative: Solve first equation for x?

2x = 32 - 3y → x = (32 - 3y)/2

Plug into second:
3*(32 - 3y)/2 + y = 10
Multiply both sides by 2:
3(32 - 3y) + 2y = 20
96 - 9y + 2y = 20
96 - 7y = 20
-7y = -76 → y = 76/7 → same as before.

So yes, answer is x = -2/7, y = 76/7

But let me re-read the problem — maybe I misread numbers?

Original:
2x + 3y = 32
3x + y = 10

Yes. So answer is fractional.

But perhaps there's a typo? Or maybe I should just accept it.

Wait — let me try plugging in integer values.

Try y=2 → 3x=8 → x=8/3 → 2*(8/3)+6 = 16/3+18/3=34/3≠32

Try y=4 → 3x=6 → x=2 → 2*2+12=16≠32

Try y=6 → 3x= -8 → x=-8/3 → 2*(-8/3)+18 = -16/3+54/3=38/3≠32

No integer solution. So fractions are fine.

But let’s keep going — maybe I’ll see a pattern.

Actually, let’s do Problem 3 and come back.

---

Problem 3:
```
12m + 10n = 35
-m + 2n = 17
```

Step 1: Solve second equation for m.
→ -m = 17 - 2n → m = 2n - 17

Step 2: Plug into first equation.
→ 12(2n - 17) + 10n = 35
→ 24n - 204 + 10n = 35
→ 34n = 239
→ n = 239/34

That’s ugly. Let me check calculation.

12*(2n - 17) = 24n - 204 ✔️
+10n → 34n - 204 = 35 → 34n = 239 → n=239/34

Then m = 2*(239/34) - 17 = 478/34 - 578/34 = -100/34 = -50/17

Check:
12*(-50/17) + 10*(239/34) = -600/17 + 2390/34 = -1200/34 + 2390/34 = 1190/34 = 35 ✔️
-(-50/17) + 2*(239/34) = 50/17 + 478/34 = 100/34 + 478/34 = 578/34 = 17 ✔️

So correct, but messy.

Perhaps the problems are designed to have nice answers? Maybe I made a mistake in reading.

Wait — let’s look at Problem 4.

---

Problem 4:
```
4x - y = 1
2x + 3y = 11
```

Step 1: Solve first equation for y.
→ y = 4x - 1

Step 2: Plug into second.
→ 2x + 3(4x - 1) = 11
→ 2x + 12x - 3 = 11
→ 14x = 14
→ x = 1

Step 3: y = 4(1) - 1 = 3

✔ Check:
4(1) - 3 = 1 ✔️
2(1) + 9 = 11 ✔️

Good!

---

Problem 5:
```
7a - 2b = 14
5a + 4b = -9
```

Step 1: Solve first equation for b? Or a? Let’s solve for b.

7a - 2b = 14 → -2b = 14 - 7a → b = (7a - 14)/2

Step 2: Plug into second.
→ 5a + 4*((7a - 14)/2) = -9
→ 5a + 2*(7a - 14) = -9
→ 5a + 14a - 28 = -9
→ 19a = 19
→ a = 1

Step 3: b = (7*1 - 14)/2 = (-7)/2 = -3.5

Check:
7(1) - 2(-3.5) = 7 + 7 = 14 ✔️
5(1) + 4(-3.5) = 5 - 14 = -9 ✔️

Good.

---

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

Step 1: Solve second equation for x.
→ 2x = -2 - 3y → x = (-2 - 3y)/2

Step 2: Plug into first.
→ -4*[(-2 - 3y)/2] + 2y = 3
→ -2*(-2 - 3y) + 2y = 3
→ 4 + 6y + 2y = 3
→ 8y = -1
→ y = -1/8

Step 3: x = (-2 - 3*(-1/8))/2 = (-2 + 3/8)/2 = (-16/8 + 3/8)/2 = (-13/8)/2 = -13/16

Check:
-4*(-13/16) + 2*(-1/8) = 52/16 - 2/8 = 13/4 - 1/4 = 12/4 = 3 ✔️
2*(-13/16) + 3*(-1/8) = -26/16 - 3/8 = -13/8 - 3/8 = -16/8 = -2 ✔️

Correct.

---

Now, going back to Problems 2 and 3 — they gave fractions, which is fine, but let me double-check if I misread the original image.

Looking back at user’s image description:

Problem 2:
2x + 3y = 32
3x + y = 10

Problem 3:
12m + 10n = 35
-m + 2n = 17

Yes, those are correct.

But perhaps in Problem 2, it’s meant to be 2x + 3y = 3 and 3x + y = 10? No, user wrote 32.

Alternatively, maybe I should present all answers as fractions.

But let’s list them clearly.

Final Answers:

1. m=4, n=-1
2. x=-2/7, y=76/7
3. m=-50/17, n=239/34
4. x=1, y=3
5. a=1, b=-7/2
6. x=-13/16, y=-1/8

But Problem 3: n=239/34, m=-50/17

Note: -50/17 = -100/34, so consistent.

Now, for the final answer, since the user didn’t specify which problem, but the image has 6 problems, I think we need to provide all solutions.

But the instruction says “solve the problem” — singular — but image has multiple. Probably expects all.

In the response, I’ll list all.

But let me write them neatly.

---

After careful calculation:

Problem 1:
m = 4, n = -1

Problem 2:
x = -2/7, y = 76/7

Problem 3:
m = -50/17, n = 239/34

Problem 4:
x = 1, y = 3

Problem 5:
a = 1, b = -7/2

Problem 6:
x = -13/16, y = -1/8

All checked.

Final Answer:
Problem 1: m=4, n=-1
Problem 2: x=-2/7, y=76/7
Problem 3: m=-50/17, n=239/34
Problem 4: x=1, y=3
Problem 5: a=1, b=-7/2
Problem 6: x=-13/16, y=-1/8