91Ó°ÊÓ

When solving linear equations, the goal is often to find the value of the variable. This means we need to 'isolate the variable'. In our example, the given equation is \(\frac{1}{3} b = -4\). \
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To isolate the variable, we need to get 'b' alone on one side of the equation. Whenever you have a fraction with a variable, a good strategy is to eliminate the fraction by multiplying both sides of the equation by the denominator of the fraction. This operation helps us get rid of the fraction and brings us closer to solving for the variable. In this exercise: \(\frac{1}{3} b\) means that one-third of 'b' equals -4. By multiplying both sides by 3, we remove the fraction: \(\frac{1}{3} b \times 3 = -4 \times 3\).

After isolating the variable, the next step is to simplify the equation. Simplifying refers to performing the operations on both sides to make the equation straightforward. In our example: \(\frac{1}{3} b \times 3\) simplifies to 'b', and \(-4 \times 3\) simplifies to -12. \
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Hence, the simplified form of the equation, after multiplying both sides by 3, is: \ b = -12 \. \
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Simplifying always involves making the equation easier to understand and solve, and it often means getting rid of any fractions or complicated terms.

Multiplying both sides of an equation by the same number is a fundamental principle in algebra that helps maintain balance. In our exercise, we have the equation: \(\frac{1}{3} b = -4\). \
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We want to isolate 'b', so we multiply both sides by 3: \(\frac{1}{3} b \times 3 = -4 \times 3\). \
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This step ensures that the equation remains balanced, because multiplying or dividing both sides of an equation by the same number doesn't change the equality. After multiplying, the left side simplifies to 'b' and the right side becomes -12, thus giving us the solution: \ b = -12 \.