- Start with the linear equation \(7r + 21 = 49r\).
- Isolate the variable term by subtracting \(7r\) from both sides to get \(21 = 42r\).
- Divide both sides by \(42\) to isolate \(r\).
- The solution is \(r = \frac{21}{42} = \boxed{0.5}\).
Explanation
- Analyzing the Equation
The given equation is \(7r + 21 = 49r\). This is a linear equation in one variable, \(r\). Our objective is to find the value of \(r\) that satisfies this equation. To do this, we need to isolate \(r\) on one side of the equation.
- Isolating the Variable Term
To isolate the variable \(r\), we start by subtracting \(7r\) from both sides of the equation to collect all terms involving \(r\) on one side. This gives us:
\[21 = 49r - 7r\]
Simplifying the expression on the right-hand side by performing the subtraction \(49r - 7r = 42r\), the equation becomes:
\[21 = 42r\]
- Solving for r
Now, we divide both sides of the equation by \(42\) to solve for \(r\):
\[r = \frac{21}{42}\]
Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, which is \(21\), we get:
\[r = \frac{1}{2}\]
Or, in decimal form, \(r = 0.5\).
Examples
Suppose you are calculating the cost of a service where a fixed fee of \$21 is added to a variable cost that is 7 times the hourly rate 'r'. If this total cost is equivalent to 49 times the hourly rate 'r', you can set up the equation 7r + 21 = 49r to determine the value of 'r'. Solving this equation helps you find the exact hourly rate required to balance these two different pricing structures.