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Q.

Find the value of x:

$(1) \frac{4 x + 7}{9 - 3 x} = \frac{1}{4}$

$(2) \frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}$

$(3) 3 (5 x - 7) - 2 (9 x - 11) = 4 (8 x - 13) - 17$

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a

x = - 1

,

x = \frac{1}{2}, x = 2

b

x = - 2

,

x = \frac{1}{2}, x = 2

c

x = - 4

,

x = \frac{1}{2}, x = 2

d

None of these

answer is A.

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Detailed Solution

Concept- We are given four equations in this problem, each with just one unknown variable. Additionally, we must solve each equation separately in order to determine the values of the variables.
Let's consider each equation individually and resolve it one at a time.
What we have for equation (1) is

\frac{4 x + 7}{9 - 3 x} = \frac{1}{4}

This can be resolved by transposing the right-side denominator to the numerator of the left-side fraction and the left-side numerator to the right-side denominator.

\Rightarrow 4 \times (4 x + 7) = 1 \times (9 - 3 x)

By removing the parentheses, we can now expand the phrases on both sides:

\Rightarrow 4 \times 4 x + 4 \times 7 = 9 - 3 x

\Rightarrow 16 x + 28 = 9 - 3 x

Now we can take the term with the variable on the other side and all the constants on the first side:

\Rightarrow 16 x + 28 = 9 - 3 x \Rightarrow 16 x + 3 x = 9 - 28

It can be further solved as follows:

19 x = - 19 .

After multiplying both sides by 19,
We obtain:

x = - 1 .

As a result, we arrived to the equation's solution,

x = - 1 .

(2) \frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}

Transposing the denominators on both sides to the other side will fix this problem.

\Rightarrow \frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}

\Rightarrow (2 - 7 x) (4 + 5 x) = (3 + 7 x) (1 - 5 x)

Now that the parenthesis are open, we may expand the expressions:

\Rightarrow 2 (4 + 5 x) - 7 x (4 + 5 x) = 3 (1 - 5 x) + 7 x (1 - 5 x)

We may further simplify it as follows:

8 + 10 x - 28 x - 35 x^{2} = 3 - 15 x + 7 x - 35 x^{2}

= 8 + 10 x - 28 x = 3 - 15 x + 7 x

We obtain the following after dividing the variable and constants on each side:

= - 18 x + 8 x = - 8 + 3

= - 10 x = - 5 .

When we multiply both sides by -10, we obtain

- 10 x = - 5

x = \frac{- 5}{- 10} = \frac{1}{2} .

Consequently, we solved the equation to get the result

x = \frac{1}{2} .

(3)The formula is

3 (5 x - 7) - 2 (9 x - 11) = 4 (8 x - 13) - 17 .

This can be fixed by expanding the expression and opening up the parenthesis:

\Rightarrow 3 \times 5 x - 3 \times 7 - 2 \times 9 x + 2 \times 11 = 4 \times 8 x - 13 \times 4 - 17

\Rightarrow 15 x - 21 - 18 x + 22 = 32 x - 52 - 1

7
Let's now divide the variables and constants between the two sides:

\Rightarrow 15 x - 18 x - 32 x = 21 - 22 - 52 - 17

\Rightarrow - 35 x = - 70

The result of multiplying both sides of the equation by

35 is : 35 x = 70

x = \frac{70}{35} = 2 .

As a result, this equation may be solved, giving us the answer x =2.
Hence, option 1 is correct.

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