Solve.
$\frac{x+9}{9}=\frac{x-1}{8}$
Answer (1)
Multiply both sides by 72 to eliminate fractions: 8 ( x + 9 ) = 9 ( x − 1 ) .
Expand both sides: 8 x + 72 = 9 x − 9 .
Isolate x by subtracting 8x and adding 9 to both sides: 72 + 9 = x .
Simplify to find the value of x: x = 81 .
Explanation
Understanding the Problem We are given the equation 9 x + 9 = 8 x − 1 and our goal is to solve for x . This involves algebraic manipulation to isolate x on one side of the equation.
Eliminating Fractions To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which are 9 and 8. The LCM of 9 and 8 is 72. Multiplying both sides by 72, we get: 72 ⋅ 9 x + 9 = 72 ⋅ 8 x − 1 Simplifying this gives: 8 ( x + 9 ) = 9 ( x − 1 )
Expanding the Equation Next, we expand both sides of the equation by distributing the numbers outside the parentheses: 8 x + 8 ( 9 ) = 9 x − 9 ( 1 ) 8 x + 72 = 9 x − 9
Isolating x Now, we want to isolate x . We can subtract 8 x from both sides of the equation: 8 x + 72 − 8 x = 9 x − 9 − 8 x 72 = x − 9
Solving for x To solve for x , we add 9 to both sides of the equation: 72 + 9 = x − 9 + 9 81 = x
Verification Therefore, the solution to the equation is x = 81 . To verify, we can substitute x = 81 back into the original equation: 9 81 + 9 = 8 81 − 1 9 90 = 8 80 10 = 10 Since the equation holds true, our solution is correct.
Final Answer The solution to the equation 9 x + 9 = 8 x − 1 is x = 81 .
Examples
Imagine you're adjusting the ingredients in a recipe to make a larger batch. If the original recipe calls for a ratio of ingredients expressed as a fraction, and you want to scale it up, you'll need to solve an equation similar to this one. For example, if you know that 9 flour + 9 = 8 sugar - 1 gives the correct ratio, solving for the amount of flour (x) ensures your larger batch maintains the same delicious balance of flavors. This type of problem is also applicable when scaling maps or architectural drawings.
