= 90 + 2ab \Rightarrow 2ab = 54 \Rightarrow ab = 27 - Redraw
Understanding the Equation: Solving 90 + 2ab = 54 Step-by-Step
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 variablesโlike 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 variablesโa crucial foundation for more advanced algebra.
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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 โ 90 = 54 โ 90
2ab = โ36
```
**Step 2: Divide both sides by 2**
Next, divide both sides by 2 to isolate `ab`:
2ab รท 2 = โ36 รท 2 Wait! At this point, we see
ab = โ18
ab = โ18, but this doesnโt match the expected value (ab = 27). This discrepancy indicates a potential typo or misstatement in the original equation.
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Fixing the Equation for a Correct Result: ab = 27
Letโs 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 ร 27 = 54
90 + 54 = 144 โ 54
``
Clearly,90 + 54 = 144โso 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:90 + 2ab = 144
What if 90 + 2ab = 90 + 54?
Then:Subtract 90:2ab = 54Divide by 2:ab = 27โ
Summary: Corrected Workflow
- Start with90 + 2ab = 144
2. Subtract 90:2ab = 543. Divide by 2:ab = 27
This confirms the logic and shows how precise algebraic manipulation leads to accurate results.
