Solve for $z$.
$\frac{z-6}{2 z+1}=10$
$z=$
Answer (1)
Multiply both sides by ( 2 z + 1 ) : z − 6 = 10 ( 2 z + 1 ) .
Expand: z − 6 = 20 z + 10 .
Rearrange: − 19 z = 16 .
Solve for z : − 19 16 .
Explanation
Understanding the Problem We are given the equation 2 z + 1 z − 6 = 10 and we need to solve for z .
Eliminating the Fraction First, we multiply both sides of the equation by ( 2 z + 1 ) to eliminate the fraction: z − 6 = 10 ( 2 z + 1 ) .
Expanding the Equation Next, we expand the right side of the equation: z − 6 = 20 z + 10 .
Rearranging the Equation Now, we rearrange the equation to isolate z terms on one side and constant terms on the other side: z − 20 z = 10 + 6 .
Simplifying the Equation Simplify both sides of the equation: − 19 z = 16 .
Solving for z Finally, divide both sides by − 19 to solve for z : z = − 19 16 = − 19 16 . Therefore, z = − 19 16 .
Examples
Imagine you're trying to figure out how much of an ingredient to add to a recipe. If the ratio of the ingredient to the total mixture is represented by an equation like the one we solved, finding the value of 'z' helps you determine the exact quantity needed to maintain the desired ratio. This type of problem is also useful in chemistry when mixing solutions or in finance when calculating investment ratios. Understanding how to solve for a variable in such equations is a fundamental skill in many practical applications.
