Q. Solve for y: \(3.4 + 5.1(y + 8) = 85\).

Q: How do I solve 3.4+5.1(y+8)=85 step-by-step?

Distribute: 5.1(y+8)=5.1y+40.8. Combine constants: 3.4+40.8=44.2, so 5.1y+44.2=85. Subtract: 5.1y=40.8. Divide: y=\frac{40.8}{5.1}=8.

Q: What common mistakes should I watch for?

Forgetting to distribute 5.1, combining nonlike terms, arithmetic errors when subtracting 44.2 from 85, or dividing by the wrong coefficient. Verify by plugging y back into the equation.

Q: How do I check my answer is correct?

Substitute y=8: 3.4+5.1(8+8)=3.4+5.1(16)=3.4+81.6=85. If both sides match, the solution is correct.

Q: How would I solve it using fractions from the start?

Convert: 3.4 = \frac{34}{10}, 5.1 = \frac{51}{10}. Solve \frac{34}{10} + \frac{51}{10}(y+8) = 85. Multiply by 10 to clear denominators, then proceed; you get the same result y=8.

Q: Is there a faster mental shortcut for this problem?

Recognize 3.4+40.8=44.2, so 5.1y=85-44.2=40.8. Then y=40.8/5.1. Notice 40.8 is exactly 8 times 5.1, so y=8.

Answer

\[
\begin{aligned}
3.4 + 5.1(y+8) &= 85 \\
3.4 + 5.1y + 40.8 &= 85 \\
5.1y + 44.2 &= 85 \\
5.1y &= 40.8 \\
y &= 8
\end{aligned}
\]

Detailed Explanation

Write the original equation.

\[3.4 + 5.1(y + 8) = 85\]
Distribute 5.1 through the parentheses. Multiply 5.1 by each term inside the parentheses.

\[5.1(y + 8) = 5.1\cdot y + 5.1\cdot 8 = 5.1y + 40.8\]

So the equation becomes

\[3.4 + 5.1y + 40.8 = 85\]
Combine like (constant) terms on the left side. Add 3.4 and 40.8.

\[3.4 + 40.8 = 44.2\]

So the equation simplifies to

\[5.1y + 44.2 = 85\]
Isolate the term with y. Subtract 44.2 from both sides.

\[5.1y = 85 – 44.2 = 40.8\]
Solve for y by dividing both sides by 5.1.

\[y = \frac{40.8}{5.1}\]

To simplify the fraction, multiply numerator and denominator by 10:

\[y = \frac{408}{51}\]

Divide: 51 times 8 equals 408, so

\[y = 8\]
Final answer.

y = 8

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FAQs

How do I solve \(3.4+5.1(y+8)=85\) step-by-step?

Distribute: \(5.1(y+8)=5.1y+40.8\). Combine constants: \(3.4+40.8=44.2\), so \(5.1y+44.2=85\). Subtract: \(5.1y=40.8\). Divide: \(y=\frac{40.8}{5.1}=8\).

Why distribute first instead of combining terms?

Distribution removes the parentheses so like terms appear. You cannot combine \(3.4\) with \(5.1(y+8)\) until you expand to \(5.1y+40.8\).

How do I handle the decimals when solving?

You can work with decimals directly or convert to fractions. Here \(40.8/5.1=408/51=136/17=8\). Multiplying numerator and denominator by 10 avoids decimal arithmetic mistakes.

What common mistakes should I watch for?

Forgetting to distribute \(5.1\), combining nonlike terms, arithmetic errors when subtracting \(44.2\) from \(85\), or dividing by the wrong coefficient. Verify by plugging \(y\) back into the equation.

How do I check my answer is correct?

Substitute \(y=8\): \(3.4+5.1(8+8)=3.4+5.1(16)=3.4+81.6=85\). If both sides match, the solution is correct.

Could there be no solution or infinitely many solutions?

How would I solve it using fractions from the start?

Convert: \(3.4 = \frac{34}{10}, 5.1 = \frac{51}{10}\). Solve \(\frac{34}{10} + \frac{51}{10}(y+8) = 85\). Multiply by 10 to clear denominators, then proceed; you get the same result \(y=8\).

Is there a faster mental shortcut for this problem?

Recognize \(3.4+40.8=44.2\), so \(5.1y=85-44.2=40.8\). Then \(y=40.8/5.1\). Notice \(40.8\) is exactly \(8\) times \(5.1\), so \(y=8\).

Work through the steps to find y, ok.
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