Video: Pack 3 β€’ Paper 1 β€’ Question 17

Pack 3 β€’ Paper 1 β€’ Question 17

04:40

Video Transcript

Find the solution to the equation π‘₯ minus seven divided by four multiplied by π‘₯ plus one minus two π‘₯ minus seven divided by three multiplied by π‘₯ plus one is equal to two.

In order to simplify any algebraic fractions, when we are adding or subtracting, we need to find the lowest common denominator. In this case, the π‘₯ plus one is in both denominators. Therefore, it is already Common. This means that, in this example, we’re only interested in the four and the three.

If we consider the fractions one-quarter and one-third, how do we find the lowest common denominator? We need to find the lowest common multiple of four and three, the lowest number that is in the four and the three times table. In this case, our answer is 12. The lowest common multiple of four and three is 12.

In order to find an equivalent fraction to a quarter, we need to consider what we’ve done to the bottom. Well, in this case, we’ve multiplied the bottom or denominator by three. Whatever you do to the bottom you must do to the top. So we need to multiply the numerator by three. One multiplied by three is equal to three. Therefore, the fraction one-quarter can also be written as three twelfths. In the same way, we can multiply the top and the bottom of the second fraction by four. This gives us four twelfths. One-third is equivalent to four twelfths.

As the denominators are now the same, we just need to subtract the numerators to get our final answer. Three minus four is equal to negative one. Therefore, our answer is negative one twelfth.

We’re now going to use the same method to find a common denominator with our algebraic fractions. Firstly, we’ll multiply the top and the bottom of the first fraction by three. We’ll multiply the top and bottom of the second fraction by four.

Multiplying the bottom of the first fraction by three gives us 12 multiplied by π‘₯ plus one. In the same way, multiplying the bottom of the second fraction by four also gives us 12 multiplied by π‘₯ plus one. We therefore have a common denominator. We also need to multiply the top of the first fraction by three and the top of the second fraction by four. As the denominators on the left-hand side are the same, we can rewrite this as a single fraction.

Our next step is to expand the brackets on the top. Three multiplied by π‘₯ is three π‘₯ and three multiplied by negative seven is negative 21. Expanding the second bracket gives us negative eight π‘₯ plus 28. This is because negative four multiplied by two π‘₯ is negative eight π‘₯ and negative four multiplied by negative seven is positive 28. Grouping the like terms on the top gives us negative five π‘₯ plus seven divided by 12 multiplied by π‘₯ plus one is equal to two.

Our next step is to multiply both sides by 12 multiplied by π‘₯ plus one. This cancels the denominator on the left-hand side. And we are left with negative five π‘₯ plus seven is equal to 24 multiplied by π‘₯ plus one. Expanding the brackets on the right-hand side gives us 24π‘₯ plus 24.

We now need to balance this equation to calculate π‘₯. Adding five π‘₯ to both sides of the equation gives us seven is equal to 29π‘₯ plus 24. Subtracting 24 from both sides gives us negative 17 is equal to 29π‘₯. And finally, dividing both sides by 29 gives us a value of π‘₯ of negative 17 over 29 or negative seventeen twenty-ninths. We could check this solution by substituting in our value of π‘₯ into the initial equation.

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