Systems of Equations Worksheets | Simultaneous Equations - Free Printable

Of course! I’ll solve each system of equations and determine whether it is consistent (has at least one solution) or inconsistent (has no solution).

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Key Concepts:

- A system is consistent if the lines intersect (one solution) or are the same line (infinitely many solutions).
- A system is inconsistent if the lines are parallel and never intersect (no solution).
- We can determine this by comparing slopes and y-intercepts, or by trying to solve the system algebraically.

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Let’s go one by one:

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1)

Equations:
`4x + 5y = 25`
`3x - y = 17`

Solve by substitution or elimination.

From second equation: `y = 3x - 17`

Substitute into first:
`4x + 5(3x - 17) = 25`
`4x + 15x - 85 = 25`
`19x = 110` → `x = 110/19`

Then `y = 3*(110/19) - 17 = 330/19 - 323/19 = 7/19`

✔ One solution → Consistent

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2)

Equations:
`13 = 8p - 6q`
`19 = 3p + 6q`

Add the two equations to eliminate q:

`13 + 19 = (8p - 6q) + (3p + 6q)`
`32 = 11p` → `p = 32/11`

Plug into one equation to find q:

`13 = 8*(32/11) - 6q`
`13 = 256/11 - 6q`
`6q = 256/11 - 143/11 = 113/11`
`q = 113/(11*6) = 113/66`

✔ One solution → Consistent

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3)

Equations:
`-28 = -10m + 18n`
`-5m + 9n = -14`

Rewrite first equation:
`-10m + 18n = -28`

Divide entire equation by 2:
`-5m + 9n = -14` ← This is identical to the second equation!

So both equations represent the same line.

✔ Infinitely many solutions → Consistent

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4)

Equations:
`5z = 3t + 6`
`8z = 10t + 3`

Write both in standard form:

Eq1: `5z - 3t = 6`
Eq2: `8z - 10t = 3`

Multiply Eq1 by 10 and Eq2 by 3 to eliminate t:

Eq1×10: `50z - 30t = 60`
Eq2×3: `24z - 30t = 9`

Subtract:
`(50z - 30t) - (24z - 30t) = 60 - 9`
`26z = 51` → `z = 51/26`

Now plug back to find t:

From Eq1: `5*(51/26) - 3t = 6`
`255/26 - 3t = 6 = 156/26`
`-3t = 156/26 - 255/26 = -99/26`
`t = 33/26`

✔ One solution → Consistent

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5)

Equations:
`12x = 16r + 5`
`-4x + 3r = 17`

Rewrite first: `12x - 16r = 5`
Second: `-4x + 3r = 17`

Multiply second equation by 3:
`-12x + 9r = 51`

Now add to first equation:

`(12x - 16r) + (-12x + 9r) = 5 + 51`
`-7r = 56` → `r = -8`

Plug into second equation:
`-4x + 3*(-8) = 17`
`-4x - 24 = 17`
`-4x = 41` → `x = -41/4`

✔ One solution → Consistent

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6)

Equations:
`c = 2d + 9`
`10d = -6 + 5c`

Substitute c from first into second:

`10d = -6 + 5*(2d + 9)`
`10d = -6 + 10d + 45`
`10d = 10d + 39`
`0 = 39` ✘

This is a contradiction — no solution.

🚫 Inconsistent

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7)

Equations:
`16h = 8c - 21`
`13c = 14h + 17`

Rewrite:

Eq1: `16h - 8c = -21`
Eq2: `-14h + 13c = 17`

Use elimination. Multiply Eq1 by 13 and Eq2 by 8:

Eq1×13: `208h - 104c = -273`
Eq2×8: `-112h + 104c = 136`

Add them:
`(208h - 104c) + (-112h + 104c) = -273 + 136`
`96h = -137` → `h = -137/96`

Now plug into Eq1 to find c:

`16*(-137/96) - 8c = -21`
`-2192/96 - 8c = -21`
Simplify: `-137/6 - 8c = -21`
`-8c = -21 + 137/6 = (-126 + 137)/6 = 11/6`
`c = -11/48`

✔ One solution → Consistent

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8)

Equations:
`18v + 9w - 22 = 0`
`6v + 3w - 35 = 0`

Rewrite:

Eq1: `18v + 9w = 22`
Eq2: `6v + 3w = 35`

Notice that Eq1 is 3×Eq2? Let’s check:

3×Eq2: `18v + 9w = 105`

But Eq1 says `18v + 9w = 22`

So we have:
`18v + 9w = 22`
`18v + 9w = 105`

This is impossible → contradiction.

🚫 Inconsistent

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9)

Equations:
`7u = -8v - 11`
`-14u + 16v = 33`

Rewrite Eq1: `7u + 8v = -11`

Multiply Eq1 by 2: `14u + 16v = -22`

Now add to Eq2:
`(14u + 16v) + (-14u + 16v) = -22 + 33`
Wait — let’s check:

Actually, Eq2 is: `-14u + 16v = 33`

So adding Eq1×2 and Eq2:

`14u + 16v = -22`
`-14u + 16v = 33`
Add: `32v = 11` → `v = 11/32`

Now plug into Eq1:
`7u + 8*(11/32) = -11`
`7u + 88/32 = -11`
`7u + 11/4 = -11`
`7u = -11 - 11/4 = -55/4`
`u = -55/28`

✔ One solution → Consistent

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10)

Equations:
`15 = -2x + 6d`
`-4x + 12d = 30`

Rewrite:

Eq1: `-2x + 6d = 15`
Eq2: `-4x + 12d = 30`

Notice Eq2 is exactly 2×Eq1:

2×Eq1: `-4x + 12d = 30` ← matches Eq2!

So both equations are the same line.

✔ Infinitely many solutions → Consistent

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## ✔ Final Answers:

1. Consistent
2. Consistent
3. Consistent
4. Consistent
5. Consistent
6. Inconsistent
7. Consistent
8. Inconsistent
9. Consistent
10. Consistent

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Let me know if you’d like to see any step-by-step graphing or slope comparison too!