Solve for $x$. Express your answer as a whole number, a decimal, or a fraction in simplest form.
$6(4 x+5)+3=2(10 x-1)$
We are given the linear equation:
\[6(4x+5)+3 = 2(10x-1)\]
The objective is to find the value of \(x\). This equation involves basic algebraic principles, such as distributing coefficients across parentheses, collecting like terms, and isolating the variable \(x\). We will proceed by simplifying both sides of the equation step by step.
First, we distribute the constants on both sides of the equation.
On the left-hand side (LHS):
\[6(4x+5) + 3 = 6 \cdot 4x + 6 \cdot 5 + 3 = 24x + 30 + 3 = 24x + 33\]
On the right-hand side (RHS):
\[2(10x-1) = 2 \cdot 10x - 2 \cdot 1 = 20x - 2\]
So, our equation becomes:
\[24x + 33 = 20x - 2\]
Next, we group all terms involving \(x\) on one side and the constant terms on the other side. We subtract \(20x\) from both sides:
\[24x - 20x + 33 = -2\]
\[4x + 33 = -2\]
Now, we subtract 33 from both sides to isolate the term with \(x\):
\[4x = -2 - 33\]
\[4x = -35\]
Finally, we divide by the coefficient of \(x\), which is 4:
\[x = \frac{-35}{4}\]
This can also be written in decimal form as \(x = -8.75\). Therefore, the solution is \(\boxed{-\frac{35}{4}}\).
Linear equations are fundamental in many real-world applications, such as budgeting or predicting costs. For example, if a company charges a fixed base fee for a service plus a variable hourly rate, an equation similar to the one solved here can determine the number of hours of service you can afford for a specific budget. By setting up the equation, you can easily isolate the variable representing hours and make informed decisions about your spending.