91Ó°ÊÓ

When faced with a linear equation such as the textbook example, the first step is always algebraic simplification. This process involves combining like terms and simplifying the equation to its more basic form. In the given exercise, we start with the equation \(8m = 3m + 20\). The goal is to combine all the terms with the variable \(m\) on one side and constants on the other.

To achieve this, we perform the same operation on both sides of the equation; we subtract \(3m\) from each side, yielding \(8m - 3m = 3m + 20 - 3m\). The reason we can do this is because of the property of equality, which states that if you do the same thing to both sides of an equation, the two sides remain equal. This simplifies to \(5m = 20\) because \(8m - 3m\) combines to \(5m\), and \(3m - 3m\) cancels out entirely. Now the equation is simpler and ready for the next step—the isolation of the variable.

After simplification, the next crucial step in solving a linear equation is the isolation of variables. This means manipulating the equation so that the variable we're solving for, in this case \(m\), stands alone on one side of the equation. From the simplified form \(5m = 20\), we can isolate \(m\) by dividing both sides by 5, which is the coefficient of \(m\).

Why do we divide? The division is the inverse operation of multiplication and is used here to 'undo' the multiplication of \(m\) by 5. Performing \(\frac{5m}{5} = \frac{20}{5}\), we simplify it to \(m = 4\). Isolating the variable is a pivotal step as it gives us the value of the variable that satisfies the original equation. It is important to perform the same operation on both sides to maintain the balance—just like a scale in perfect equilibrium.

Finally, solution verification is the litmus test for the accuracy of the solution we've obtained. In our example, we have found that \(m = 4\) is the solution. To verify, we substitute this value back into the original equation and check to see if the equation holds true. The original equation was \(8m = 3m + 20\), so with the substitution we get \(8(4) = 3(4) + 20\).

Carrying out the multiplications, we have \(32 = 12 + 20\). The right side simplifies further to \(32 = 32\), showing that both sides of the equation are indeed equal when \(m = 4\). This tells us that our solution is correct. It's crucial to verify solutions to prevent any errors in the problem-solving process. If the numbers on both sides of the equation match, as they do in this case, it confirms that you've correctly solved the equation.