Solve for x ax+2c=bx+3d

Solution:

Step 1:

Eliminate $bx$ from both sides: $ax+2c-bx=3d$.

Step 2:

Remove $2c$ from both sides to isolate terms with $x$: $ax-bx=3d-2c$.

Step 3:

Extract the common variable $x$ from the terms $ax$ and $-bx$.

Step 3.1:

Take $x$ out of $ax$: $x(a-bx)=3d-2c$.

Step 3.2:

Extract $x$ from $-bx$: $xa-x(b)=3d-2c$.

Step 3.3:

Combine the factored terms: $x(a-b)=3d-2c$.

Step 4:

Divide the equation by $a-b$ to solve for $x$.

Step 4.1:

Apply the division to both terms: $\frac{x(a-b)}{a-b}=\frac{3d-2c}{a-b}$.

Step 4.2:

Simplify the equation by canceling out terms.

Step 4.2.1:

Identify and cancel the common factors.

Step 4.2.1.1:

Cancel $a-b$: $\frac{x\cancel{(a-b)}}{\cancel{(a-b)}}=\frac{3d}{a-b}-\frac{2c}{a-b}$.

Step 4.2.1.2:

Simplify to find $x$: $x=\frac{3d}{a-b}-\frac{2c}{a-b}$.

Step 4.3:

Combine the terms on the right-hand side.

Step 4.3.1:

Rearrange the negative sign: $x=\frac{3d}{a-b}-\frac{2c}{a-b}$.

Step 4.3.2:

Merge the fractions with a common denominator: $x=\frac{3d-2c}{a-b}$.

Knowledge Notes:

The problem-solving process involves algebraic manipulation to solve for the variable $x$ in the linear equation $ax+2c=bx+3d$. The steps include:

1. Rearranging the equation to group like terms.

2. Factoring out the common variable.

3. Simplifying the equation by canceling out common factors.

4. Combining like terms to reach the solution.

Key algebraic concepts used in this process include:

• Combining like terms: This is the process of adding or subtracting terms that have the same variable raised to the same power.

• Factoring: This involves taking out a common factor from terms to simplify expressions.

• Simplifying expressions: This includes canceling out common factors in a fraction and combining fractions with a common denominator.

• Solving linear equations: The goal is to isolate the variable on one side of the equation to find its value. This often requires the use of inverse operations, such as adding or subtracting terms on both sides, multiplying or dividing both sides by a number, etc.

In this particular problem, the equation is first rearranged to group the terms with $x$ on one side and the constant terms on the other. Then, $x$ is factored out to simplify the left side of the equation. Finally, the equation is divided by the coefficient of $x$ to solve for $x$. The solution is expressed as a single fraction with a common denominator.