Question - Solving an Equation with Incomplete Image Clarity
Solution:
The image is not entirely clear, but I will provide the solution to the equation based on what is visible:Given: \[ \frac{1}{a} + \frac{1}{b} = \frac{1}{4.5} \]First, find the common denominator for $$\frac{1}{a}$$ and $$\frac{1}{b}$$, which is $$ab$$. \[ \frac{b}{ab} + \frac{a}{ab} = \frac{1}{4.5} \]\[ \frac{a + b}{ab} = \frac{1}{4.5} \]Cross-multiply to solve for $$a + b$$:\[ a + b = ab \cdot \frac{1}{4.5} \]Convert 4.5 to a fraction:\[ a + b = ab \cdot \frac{1}{\frac{9}{2}} \]Inverse the fraction:\[ a + b = ab \cdot \frac{2}{9} \]Divide both sides by ab to solve for $$\frac{1}{ab}$$:\[ \frac{a+b}{ab} = \frac{2}{9} \]Therefore:\[ \frac{1}{ab} = \frac{2}{9(a+b)} \]The correct expression that represents $$\frac{1}{ab}$$ is:\[ \frac{2}{9(a+b)} \]Looking at the options provided, the answer is:b) $$\frac{2}{9(a+b)}$$
Tutor Solution:
I see the image contains a math problem that looks like a fraction equation. The equation is:$$ \frac{a}{4.5} + \frac{a}{x} = \frac{1}{2.5} $$The task is to solve for $$ x $$. Let's take this step by step:1. Find a common denominator for the fractions on the left side of the equation. The common denominator for $$ 4.5 $$ and $$ x $$ will be $$ 4.5x $$.2. Express each fraction with the common denominator. $$ \frac{a}{4.5} $$ becomes $$ \frac{ax}{4.5x} $$ and $$ \frac{a}{x} $$ becomes $$ \frac{4.5a}{4.5x} $$.3. Rewrite the equation with these new fractions: $$ \frac{ax}{4.5x} + \frac{4.5a}{4.5x} = \frac{1}{2.5} $$.4. Combine the terms on the left side: $$ \frac{ax + 4.5a}{4.5x} = \frac{1}{2.5} $$.5. Since we have a common denominator on the left, we can add the numerators directly: $$ \frac{ax + 4.5a}{4.5x} $$ simplifies to $$ \frac{a(x + 4.5)}{4.5x} $$.6. Now, we can cross-multiply to get rid of the fractions: $$ a(x + 4.5) = \frac{1}{2.5} * 4.5x $$.7. Simplify the right side by multiplying $$ \frac{1}{2.5} * 4.5x $$: Which simplifies down to $$ 1.8x $$ because $$ \frac{4.5}{2.5} = 1.8 $$.8. So we have: $$ a(x + 4.5) = 1.8x $$.9. To solve for $$ x $$, you want to first distribute the $$ a $$ on the left side: $$ ax + 4.5a = 1.8x $$.10. Now, as we're solving for $$ x $$, we need to isolate $$ x $$ and its coefficients on one side while moving constants to the other side. $$ ax - 1.8x = -4.5a $$.11. Factor out the $$ x $$ from the left side: $$ x(a - 1.8) = -4.5a $$.12. To isolate $$ x $$ fully, divide both sides of the equation by $$ (a - 1.8) $$: $$ x = \frac{-4.5a}{a - 1.8} $$.Now we have the value of $$ x $$ in terms of $$ a $$. Unfortunately, the options provided in the image are not clearly visible, so I cannot confirm which option matches with our solution. If you could clarify the options or make the image clearer, I could help identify the correct one.
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