Understanding the Equation: Solving 90 + 2ab = 54 Step-by-Step

In algebra, solving equations step-by-step is essential for clarity and accuracy. One such common problem involves simplifying linear equations with variables—like solving 90 + 2ab = 54 to find the product ab = 27. This article breaks down the process clearly, helping students, tutors, and math learners master basic algebraic manipulation.

Understanding the Context

What Is the Equation?

We begin with the equation:
90 + 2ab = 54

Our goal is to isolate ab and find its value. This type of equation often appears in math education when teaching simplification and solving for variables—a crucial foundation for more advanced algebra.

= 90 + 2ab \Rightarrow 2ab = 54 \Rightarrow ab = 27

Key Insights

Step-by-Step Solution

Step 1: Subtract 90 from both sides
To eliminate the constant term on the left, subtract 90 from both sides: 

90 + 2ab – 90 = 54 – 90  
2ab = –36
```

**Step 2: Divide both sides by 2**  
Next, divide both sides by 2 to isolate `ab`:

2ab ÷ 2 = –36 ÷ 2
ab = –18

Wait! At this point, we see ab = –18, but this doesn’t match the expected value (ab = 27). This discrepancy indicates a potential typo or misstatement in the original equation.

Continue Reading

But this contradicts $2a = 14$. So let’s double-check the assumption.
Actually, the second and fourth terms: $(a - d) + (a + d) = 2a = 10 \Rightarrow a = 5$
So from the first condition: first + fifth = $2a = 14 \Rightarrow a = 7$

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Final Thoughts

Fixing the Equation for a Correct Result: ab = 27

Let’s reverse-engineer the problem to achieve ab = 27. Starting again from the original equation:
90 + 2ab = 54, but if we want ab = 27, rearrange the equation correctly:
90 + 2(27) = ?
2ab = 2 × 27 = 54
90 + 54 = 144 ≠ 54
``

Clearly,90 + 54 = 144—so the original equation90 + 2ab = 54cannot yieldab = 27.

Corrected Equation for ab = 27

To satisfy the conditionab = 27, the original equation must be adjusted. Suppose instead:
What if 90 + 2ab = 90 + 54?
Then:90 + 2ab = 144Subtract 90:2ab = 54Divide by 2:ab = 27✅

Summary: Corrected Workflow

  1. Start with90 + 2ab = 1442. Subtract 90:2ab = 543. Divide by 2:ab = 27

This confirms the logic and shows how precise algebraic manipulation leads to accurate results.