Question

Solve the equation: \[\dfrac{{3k + 5}}{{4k - 3}} = \dfrac{4}{9}\]

Answer 1

Hint:
Here we need to find the value of the variable. We will use the cross multiplication method here to simplify the fraction. Then we will add or subtract the like terms, which will form a linear equation including that one variable. Then we will simplify the obtained equations further to get the required value of the variable.

Complete step by step solution:
Here, we need to solve the given equation.
\[\dfrac{{3k + 5}}{{4k - 3}} = \dfrac{4}{9}\]
Now, we will apply the cross multiplication method here to simplify these fractions. Therefore, we get
\[ \Rightarrow 9\left( {3k + 5} \right) = 4\left( {4k - 3} \right)\]
Now, we will use the distributive property of multiplication to multiply the terms of both the sides of the equation.
\[ \Rightarrow 9 \times 3k + 9 \times 5 = 4 \times 4k - 4 \times 3\]
On multiplying all the terms, we get
\[ \Rightarrow 27k + 45 = 16k - 12\]
On subtracting the term \[16k\] from both the sides, we get
\[\begin{array}{l} \Rightarrow 27k - 16k + 45 = 16k - 12 - 16k\\ \Rightarrow 11k + 45 = - 12\end{array}\]
Now, we will subtract the number 45 from both sides. Therefore, we get
\[\begin{array}{l} \Rightarrow 11k + 45 - 45 = - 12 - 45\\ \Rightarrow 11k = - 57\end{array}\]
Now, we will divide both sides of the equation by the number 11.

\[ \Rightarrow k = \dfrac{{ - 57}}{{11}}\]
Hence, this is the required solution to the given equation.

Note:
We have used cross multiplication to solve the problem. In cross-multiplication, we multiply the denominator of the second fraction with the numerator of the first fraction and the denominator of the first fraction with the numerator of the second fraction. Also we have used distributive property to simplify the terms. The distributive property of multiplication states that if \[a\] , \[b\] and \[c\] are any numbers then according to this property, \[\left( {a + b} \right)c = a \cdot c + b \cdot c\] . In this problem, there is only one variable present here and so, we have got the linear equation in one variable after cross multiplication. Linear equations are the equations which have the highest degree of 1.