91Ó°ÊÓ

The Distributive Property is a key concept in solving linear equations. It allows you to simplify expressions by distributing a single term across terms inside parentheses. In the exercise, we see this with the term \(0.5(16m + 34)\). To distribute, you multiply \(0.5\) by each term inside the parentheses separately:
\ \[0.5 \times 16m + 0.5 \times 34 = 8m + 17\] \
Note how each term inside the parentheses is individually multiplied by \(0.5\). This technique simplifies the expression, making it easier to solve the equation. The resultant expression \(8m + 17 = -15\) can then be tackled with further algebraic steps. The distributive property is particularly helpful when dealing with constants or coefficients that are outside the parentheses.

Isolating variables is a crucial step in solving linear equations. The goal is to get the variable (in this case, \(m\)) by itself on one side of the equation. Starting from the equation obtained after applying the distributive property:
\ \[8m + 17 = -15\] \
We need to eliminate the constant term on the same side as the variable. To do this, subtract 17 from both sides:
\ \[8m + 17 - 17 = -15 - 17\] \
This simplifies to:
\ \[8m = -32\] \
By performing the same operation on both sides, we keep the equation balanced and effectively isolate the term with the variable. At this point, the expression is simplified enough to solve for the variable directly.

The final step in solving a linear equation is solving for the variable. After isolating the variable term, we often need to perform one final operation to find the value of the variable. From our previous step, we have:
\ \[8m = -32\] \
To solve for \(m\), we divide both sides of the equation by the coefficient of \(m\) (which is 8 in this case):
\ \[\frac{8m}{8} = \frac{-32}{8}\] \
This simplifies to:
\ \[m = -4\] \
By dividing each side by the same number, we ensure that the equation remains balanced. Now, we have found that \(m = -4\). This step concludes the process of solving the linear equation. Always double-check your work to verify that the value of the variable satisfies the original equation.