Question

Solve the equation \(a(x+b)=b x+c\) for \(x\) if \(a \neq b\).

Short answer

The solution to the equation \(a(x+b) = bx+c\) is: \(x = \frac{c - ab}{a - b}\)

Step-by-step solution

Expand the equation

Using the distributive property, we will multiply \(a\) with both terms inside the parentheses: \(a(x+b) = ax + ab\). So the equation becomes: \[ax + ab = bx + c\]

Rearrange the equation

We will now move all the terms containing \(x\) to the left side and all the constant terms (not containing \(x\)) to the right side: \[ ax - bx = c - ab \]

Isolate x

Now, we can factor out \(x\) from the terms on the left side and then divide both sides of the equation by the resulting coefficient of \(x\). This will give us the value of \(x\): \[ (x)(a - b) = c - ab \] \[ x = \frac{c - ab}{a - b} \] So, the solution to the equation \(a(x+b) = bx+c\) is: \[x = \frac{c - ab}{a - b}\]

Key concepts

Distributive Property

Understanding the distributive property is crucial in solving linear equations like \(a(x+b)=b x+c\). This property allows us to simplify expressions by distributing a multiplier across terms within parentheses.
For example, when we have \(a(x+b)\), the distributive property lets us multiply \(a\) by both \(x\) and \(b\), resulting in \(ax + ab\).
This transformation is key to simplifying the equation and making it easier to manipulate and solve. Mastering this property can significantly speed up your ability to handle and simplify equations effectively.

Rearranging Equations

Rearranging equations is a technique that helps make equations more manageable. It involves moving terms from one side of the equation to the other.
This is often done to separate variables from constants or to group similar terms together for easier calculation.
In our example, after applying the distributive property in \(ax + ab = bx + c\), we shift terms involving \(x\) to one side \((ax - bx)\) and constants \((c - ab)\) to the other.
This sets the stage for straightforward isolation of the variable, preparing us to finalize the solution.

Isolating Variables

Isolating a variable means getting the variable alone on one side of the equation, essentially solving for that variable.
In our equation \( ax - bx = c - ab \), we aim to isolate \(x\) by first factoring it out.
We notice that \(x\) appears in both \(ax\) and \(bx\). By factoring \(x\) out, we obtain \((x)(a - b)\).
This isolates \(x\) since it appears as a single entity multiplied by another expression.
We then divide both sides by \(a - b\) to completely isolate \(x\), resulting in the solution:

• \[ x = \frac{c - ab}{a - b} \]

This step-by-step isolation not only solves the equation but also provides clarity on how variables interact in the equation.

Factoring

Factoring is a method used to simplify equations and expressions by expressing them as a product of simpler ones.
In this context, after we have rearranged and combined terms, we want to factor out terms to isolate the variable.
Observing the equation \( ax - bx = c - ab \), we can factor \(x\) out from \(ax - bx\).
By doing this, we express it as \((x)(a - b) = c - ab\), which simplifies isolating \(x\) since it is now neatly separated.
Factoring is a powerful tool that not only simplifies solving equations but also helps in understanding the structure and relationships of algebraic expressions more deeply.