What value of $x$ makes the equation shown true?
$39-.5 x=.5(9 x+8)$
We are given the equation \(39 - 0.5x = 0.5(9x + 8)\). To solve for \(x\), we first look at the right-hand side of the equation, which contains a term in parentheses multiplied by \(0.5\). We can distribute the \(0.5\) to each term inside the parentheses. Performing the multiplication, we have \(0.5 \times 9x = 4.5x\) and \(0.5 \times 8 = 4\). Therefore, the right-hand side of the equation simplifies to \(4.5x + 4\).
Now that we have distributed the terms, we rewrite the equation as:
\[39 - 0.5x = 4.5x + 4.\]
Our goal is to collect all terms containing \(x\) on one side of the equation and all constant terms on the other. We can add \(0.5x\) to both sides of the equation to isolate the \(x\) terms on the right:
\[39 = 4.5x + 0.5x + 4.\]
Combining the terms with \(x\), we get:
\[39 = 5x + 4.\]
Next, we isolate the \(5x\) term by subtracting \(4\) from both sides of the equation:
\[39 - 4 = 5x.\]
\[35 = 5x.\]
To solve for \(x\), we divide both sides of the equation by \(5\):
\[x = \frac{35}{5}.\]
\[x = 7.\]
Imagine you have a budget of \$39 for a craft project. You need to buy a specific type of fabric that costs \$0.50 per unit of length \(x\). If you also need to purchase decorative items that cost \$0.50 each, and you buy 9 units of fabric and 8 decorative items in a specific arrangement, the cost can be modeled by an equation like the one we solved. By finding the value of @@MATH_TOKEN_32@@, you determine exactly how much fabric you can afford to stay within your budget of \$39. Algebra like this helps in managing finances and resources effectively in real-world projects.