Solve the equation $-12.2 z-13.4 z=-179.2$
The given equation is \(-12.2z - 13.4z = -179.2\). This is a linear equation in one variable \(z\). To solve for \(z\), we must first combine the like terms on the left-hand side. Both terms contain \(z\), so we can factor out \(z\), resulting in \((-12.2 - 13.4)z = -179.2\).
Next, we calculate the sum of the coefficients of \(z\): \(-12.2 - 13.4 = -25.6\). Substituting this back into the equation, we obtain: \[-25.6z = -179.2.\]
To isolate \(z\), we divide both sides of the equation by \(-25.6\): \[z = \frac{-179.2}{-25.6}.\] Carrying out this division, we find the value of \(z\): \[z = 7.\]
Linear equations are fundamental in everyday calculations. For example, if you are buying items that cost a certain amount and you know the total cost, you use a linear equation to find the number of items. Suppose you purchase multiple items at a price of \(\$12.20\) and \(\$13.40\) per unit respectively, and your total bill is \(\$179.20\), you can represent the total quantity of each item as \(z\), leading to the equation \(-12.2z - 13.4z = -179.2\) to determine the unit distribution or quantity value. Solving such problems helps in budgeting and resource management.