We are given the reciprocal equation one over p plus one over q equals one over seven. Our goal is to solve for q in terms of p. This type of equation appears in physics, such as the lensmaker's formula.
To solve for q, we first isolate the term containing q. We do this by subtracting one over p from both sides of the equation. This gives us one over q equals one over seven minus one over p.
Now we need to combine the fractions on the right side. To do this, we find a common denominator, which is seven p. We rewrite one over seven as p over seven p, and one over p as seven over seven p. Subtracting these gives us p minus seven over seven p.
Since we have one over q equals p minus seven over seven p, we can take the reciprocal of both sides to solve for q. This gives us q equals seven p over p minus seven. We must remember that p cannot equal seven, as this would make the denominator zero.
Let's verify our solution with specific examples. First, if p equals fourteen, then q equals seven times fourteen divided by fourteen minus seven, which is ninety-eight divided by seven, giving us fourteen. Checking: one over fourteen plus one over fourteen equals two over fourteen, which simplifies to one over seven. Correct! Second, if p equals twenty-one, then q equals seven times twenty-one divided by twenty-one minus seven, which is one hundred forty-seven divided by fourteen, giving us ten point five. Checking: one over twenty-one plus one over ten point five equals one over twenty-one plus two over twenty-one, which is three over twenty-one, or one over seven. Correct again!
Finally, let's consider the domain restrictions for our solution q equals seven p over p minus seven. The solution is undefined when p equals zero or when p equals seven, as these make denominators zero in the original equation or the solution. When p is greater than seven, q is positive. When p is between zero and seven, q is negative. When p is negative, q is also positive. These restrictions are important for any physical application of this equation.