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# How do you solve for x in $ax+b=cx-d$?

Last updated date: 14th Jun 2024
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Hint: The equation given in the above question is a linear equation in one variable, that is x. The question says that we have to solve the given equation in x. In other words, we have to find the values of x which will satisfy the given equation. Try to find the value of x by performing some mathematical operations.

Complete step-by-step solution:
The given equation says that $ax+b=cx-d$. Let us analyse the given equation and check what we can do to find the value of x that satisfies the given equation. The first thing that we can do is to club the terms having x on one side of the equation (say on the left hand side of the equation).

Then, after doing this we can write the equation as $ax-cx+b=-d$.
Now, we can club the constant terms on the other side (i.e. right hand side of the equation).
With this, the above equation can be written as $ax-cx=b-d$.
Now, we can take ‘x’ as the common term from the left hand side of the equation.
Therefore, the equation can be written as $x(a-c)=b-d$.
Then,
$\Rightarrow x=\dfrac{b-d}{a-c}$
Therefore, the solution of the given equation is $x=\dfrac{b-d}{a-c}$ or we can also say that the value $x=\dfrac{b-d}{a-c}$ satisfies the given equation.

Note: There are many ways in which we can solve this question. We can also directionally add the two fractions on the left hand side of the equation and calculate the value of x. However, students must note that the number of solutions to a given equation in one variable is always less than or equal to the degree of the polynomial in the given equation.