Students have used reciprocals when thinking through relationships like 10÷2=10.1/2. They have used reciprocals to make sense of dividing with fractions. These understandings can also be applied to more complex problem-solving.
Solve The Reciprocal Equation
A Slight Twist on Equation Solving
Reciprocals are slightly different approaches to multiplying both sides of an equation by the same value. Multiplying by a reciprocal always results in a value of 1. So multiplying by a reciprocal can be used to change the coefficient in an equation into 1. Since 1x is the same as x, multiplying by the reciprocal isolates the variable.
Multiplying by the reciprocal can be applied to a one-step equation.
(3/4)x=6
Once we determine the reciprocal, familiar problem-solving steps can be used. Whatever we do on the left side of the equation, we must also do it on the right side of the equation. In this case, we are multiplying both sides by 4/3 since 4/3 is reciprocal to 3/4.
Hidden Reciprocals
Although this equation is essentially the same as the previous one, it is more difficult to recognize that it can be solved by multiplying by the reciprocal. Instead, students might multiply both sides by 5 and then solve for 3x by dividing both sides by 3.
3x/5=12
The equation can also be solved by multiplying by a reciprocal. Help students recognize that the numerator is 3 times x and the denominator is 5.
3x/5=12 is the same as (3/5). (x/1)=12
Multiplying both sides by 5/3, which is the reciprocal of ⅗, will isolate the variable.
More Complex Equations
Most students would probably groan at this problem, thinking they need to distribute the ⅖, and knowing that they will end up with fractional amounts.
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2/5(x-4)=4
Instead, we can think of ⅖ as one of the terms that is being multiplied. If we multiply by the reciprocal, which is 5/2, the ⅖ term will become 1, and will essentially be eliminated.
Multiplying by the reciprocal becomes a new tool that students can add to familiar equation-solving techniques. They should develop confidence about what types of equations will be easier to solve when the concept is applied. Ideally, it will become an approach they use confidently and can use to make sense of future concepts.
Also read: Dividing Fractions – Dividing vs Multiplying By The Reciprocal
Frequently Asked Questions on Solving Reciprocals in Equations
What does reciprocal mean?
The word reciprocal here means the inverse of either a number or a value.
What are reciprocal equations?
Reciprocal equations refer to those equations that have coefficients on one end and the opposite sign to the coefficient on the other end of the equation.
Where can I find worksheets on solving reciprocal in equations?
You can find worksheets on solving reciprocal equations and many other exclusive math resources, here.
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