Solve the equation:
$-5(x-3)-2x-2=-43$
The given problem requires solving the linear equation \(-5(x - 3) - 2x - 2 = -43\) for the variable \(x\). This equation involves the distributive property of multiplication over subtraction, combining like terms, and basic arithmetic operations on real numbers. Our goal is to isolate \(x\) to find its value.
First, we apply the distributive property to the expression \(-5(x - 3)\). This gives us \(-5 \times x - 5 \times (-3) = -5x + 15\). Substituting this back into the equation, we obtain:
\[-5x + 15 - 2x - 2 = -43.\]
Next, we group the variable terms and the constant terms on the left side of the equation. We have \(-5x\) and \(-2x\), which combine to give \(-7x\), and we have constants \(15\) and \(-2\), which combine to give \(13\):
\[-7x + 13 = -43.\]
To isolate the term with \(x\), we subtract \(13\) from both sides of the equation:
\[-7x = -43 - 13,\]
\[-7x = -56.\]
Finally, we solve for \(x\) by dividing both sides of the equation by the coefficient of \(x\), which is \(-7\):
\[x = \frac{-56}{-7},\]
\[x = 8.\]
Verifying the result: \(-5(8 - 3) - 2(8) - 2 = -5(5) - 16 - 2 = -25 - 18 = -43\), which confirms our answer is correct. Thus, the value of \(x\) is \(\boxed{8}\).
Linear equations like this are frequently used in budget planning. For example, if you have an initial amount of money and you make recurring monthly payments or deductions (like rent or subscription services), you can use a linear equation to determine how long it will take to reach a certain account balance or to find out how much you were spending per month given your final balance. This foundational algebra skill allows for clear financial tracking and informed decision-making.