Solving linear equations with two fractions - Variation Theory - Free Printable

Let’s solve each equation one by one, step by step. We’ll check if the given answer is correct or not.

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Equation 1:
\[
\frac{x + 2}{2} + \frac{x + 4}{3} = 4
\]
Given: \( x = 2 \)

Plug in \( x = 2 \):
Left side:
\[
\frac{2 + 2}{2} + \frac{2 + 4}{3} = \frac{4}{2} + \frac{6}{3} = 2 + 2 = 4
\]
Right side: 4 → ✔ Correct!

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Equation 2:
\[
\frac{x - 2}{6} - \frac{x - 4}{8} = 9
\]
Given: \( x = 212 \)

Plug in \( x = 212 \):
First term: \( \frac{212 - 2}{6} = \frac{210}{6} = 35 \)

Second term: \( \frac{212 - 4}{8} = \frac{208}{8} = 26 \)

So: \( 35 - 26 = 9 \) → ✔ Correct!

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Equation 3:
\[
\frac{x + 2}{2} + \frac{x + 4}{8} = 4
\]
Given: \( x = 4 \)

Plug in \( x = 4 \):
First term: \( \frac{4 + 2}{2} = \frac{6}{2} = 3 \)

Second term: \( \frac{4 + 4}{8} = \frac{8}{8} = 1 \)

Sum: \( 3 + 1 = 4 \) → ✔ Correct!

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Equation 4:
\[
\frac{2x - 2}{6} - \frac{2x - 4}{8} = 9
\]
Given: \( x = 106 \)

Plug in \( x = 106 \):

First term: \( \frac{2(106) - 2}{6} = \frac{212 - 2}{6} = \frac{210}{6} = 35 \)

Second term: \( \frac{2(106) - 4}{8} = \frac{212 - 4}{8} = \frac{208}{8} = 26 \)

So: \( 35 - 26 = 9 \) → ✔ Correct!

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Equation 5:
\[
\frac{x + 2}{2} + \frac{x + 4}{8} = 9
\]
Given: \( x = 12 \)

Plug in \( x = 12 \):

First term: \( \frac{12 + 2}{2} = \frac{14}{2} = 7 \)

Second term: \( \frac{12 + 4}{8} = \frac{16}{8} = 2 \)

Sum: \( 7 + 2 = 9 \) → ✔ Correct!

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Equation 6:
\[
\frac{4x - 2}{6} - \frac{4x - 4}{8} = 9
\]
Given: \( x = 53 \)

Plug in \( x = 53 \):

First term: \( \frac{4(53) - 2}{6} = \frac{212 - 2}{6} = \frac{210}{6} = 35 \)

Second term: \( \frac{4(53) - 4}{8} = \frac{212 - 4}{8} = \frac{208}{8} = 26 \)

So: \( 35 - 26 = 9 \) → ✔ Correct!

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Equation 7:
\[
\frac{x + 2}{6} + \frac{x + 4}{8} = 9
\]
Given: \( x = 28 \)

Plug in \( x = 28 \):

First term: \( \frac{28 + 2}{6} = \frac{30}{6} = 5 \)

Second term: \( \frac{28 + 4}{8} = \frac{32}{8} = 4 \)

Sum: \( 5 + 4 = 9 \) → ✔ Correct!

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Equation 8:
\[
\frac{4x - 4}{8} - \frac{4x - 2}{6} = 9
\]
Given: \( x = -55 \)

Plug in \( x = -55 \):

First term: \( \frac{4(-55) - 4}{8} = \frac{-220 - 4}{8} = \frac{-224}{8} = -28 \)

Second term: \( \frac{4(-55) - 2}{6} = \frac{-220 - 2}{6} = \frac{-222}{6} = -37 \)

Now: \( -28 - (-37) = -28 + 37 = 9 \) → ✔ Correct!

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Equation 9:
\[
\frac{x + 2}{6} - \frac{x + 4}{8} = 9
\]
Given: \( x = 220 \)

Plug in \( x = 220 \):

First term: \( \frac{220 + 2}{6} = \frac{222}{6} = 37 \)

Second term: \( \frac{220 + 4}{8} = \frac{224}{8} = 28 \)

So: \( 37 - 28 = 9 \) → ✔ Correct!

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Equation 10:
\[
\frac{4x - 4}{8} - \frac{4x - 2}{6} = 1
\]
Given: \( x = -7 \)

Plug in \( x = -7 \):

First term: \( \frac{4(-7) - 4}{8} = \frac{-28 - 4}{8} = \frac{-32}{8} = -4 \)

Second term: \( \frac{4(-7) - 2}{6} = \frac{-28 - 2}{6} = \frac{-30}{6} = -5 \)

Now: \( -4 - (-5) = -4 + 5 = 1 \) → ✔ Correct!

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Equation 11:
\[
\frac{x + 2}{6} - \frac{x - 4}{8} = 9
\]
Given: \( x = 196 \)

Plug in \( x = 196 \):

First term: \( \frac{196 + 2}{6} = \frac{198}{6} = 33 \)

Second term: \( \frac{196 - 4}{8} = \frac{192}{8} = 24 \)

So: \( 33 - 24 = 9 \) → ✔ Correct!

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Equation 12:
\[
\frac{4x - 4}{8} - \frac{4x - 2}{6} = \frac{1}{2}
\]
Given: \( x = -4 \)

Plug in \( x = -4 \):

First term: \( \frac{4(-4) - 4}{8} = \frac{-16 - 4}{8} = \frac{-20}{8} = -2.5 \)

Second term: \( \frac{4(-4) - 2}{6} = \frac{-16 - 2}{6} = \frac{-18}{6} = -3 \)

Now: \( -2.5 - (-3) = -2.5 + 3 = 0.5 = \frac{1}{2} \) → ✔ Correct!

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All answers are correct! The student did a great job solving all these equations.

Final Answer: All given solutions are correct.