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Skip to 0 minutes and 11 secondsHello. Welcome to Week 4. Let us consider the introduction, types of equations step. And let us start with the first exercise. It's a problem about cherries. Precisely, the question is, how many cherries were on the table if, after we ate 4/5 of those, we remain with three times the initial amount minus 42? This problem can be easily solved with an equation. Let us see. Call x the initial number of cherries on the table.

Skip to 1 minute and 12 secondsThen the exercise gives us some information. What do we know? That after we ate 4/5 of those, we remain with three times the initial amount minus 42. Ok then, this is the initial amount. After we have eaten the 4/5 of those, we remain with the initial amount minus the 4/5 of the initial amount. And we know, from the text of the exercise, that this is equal to three times the initial amount minus 42.

Skip to 2 minutes and 2 secondsThis is just a translation of the problem in an equation. And what do you get? This is a linear equation. Very easy to solve, but let us see. We get x minus 4/5 x minus 3 times x equal to minus 42. And then let us collect together the coefficients of x. And we have 5 over 5 which is 1 minus 4 over 5 minus 3 times 5 over 5, x equal to minus 42. And this is exactly, we have 5 as common denominator. And you have 5 minus 4, which is 1, minus 15. 1 minus 15 is minus 14.

Skip to 3 minutes and 18 secondsThen we have that this fraction has to be equal to minus -- times x has to be equal to minus 42.

Skip to 3 minutes and 30 secondsThen, now we can multiply on both sides by minus 1. And we get 14 over 5x equal to 42. We can divide on both sides by 14. Indeed, divide by 14 is exactly like to divide by 7, and then divide by 2. If we divide by 7 42, we get 6. And then we divide by 2, we get 3.

Skip to 4 minutes and 14 secondsTherefore, we remain with the equation x equal, you multiply it by 5 on both sides, x equal to 15. Therefore, the initial amount of cherries was 15. Thank you.

Types of equations in practice - Part 1

The following exercises are solved in this step.

We invite you to try to solve them before watching the video.

In any case, you will find below a PDF file with the solutions.

Exercise 1.

How many cherries were on the table if, after we ate \(4/5\) of them, we are left with 3 times the initial amount minus 42?

Exercise 2. [Solved only in the PDF file]

The sum of two consecutive integer numbers is equal to 93. What are the two numbers?

Exercise 3. [Solved only in the PDF file]

The product of two consecutive even integer numbers is \(48\). What are the two numbers?

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Precalculus: the Mathematics of Numbers, Functions and Equations

University of Padova

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