Problem 13 - Technical Mathematics with Calculus

Question

Solve for \(x .\) Try some by calculator. $$a x-b x=c+d x-m$$

Step-by-Step Solution

Answer

The solution for x is given by \(x = \frac{c+d+m}{a-b-d}\).

Step 1: Combine like terms

Combine the terms involving x on the left side of the equation and move the constant terms to the right side of the equation.

Step 2: Isolate the variable

Combine the terms with x on the left side and the constants on the right side to isolate x.

Step 3: Solve the equation

Divide both sides by the coefficient of x to find the solution for x.

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps is to simplify each side of the equation by combining like terms. Like terms are terms that have the same variables raised to the same power. In the equation, \( a x - b x = c + d x - m \), we have like terms that can be combined: the terms involving \( x \).

Here's how you can approach it:

• Identify like terms on each side of the equation.
• Add or subtract these terms as appropriate.
• For instance, \( a x \) and \( - b x \) on the left can be combined because they both contain the variable \( x \).
• Also, \( d x \) on the right side can be moved to the left side to be combined with the other \( x \) terms by adding \( -d x \) to both sides.

Once combined, these terms simplify the equation, making it easier to solve.

Isolating the Variable

The next crucial step is isolating the variable on one side of the equation to solve for it. In our equation, after combining like terms, you want to collect all the \( x \) terms on one side and all the constant terms on the other side. This helps to clear the path toward finding the value of \( x \).

To isolate the variable, you should:

• Get all terms with \( x \) on one side by using addition or subtraction.
• Move all constant terms without \( x \) to the opposite side in the same way.
• Keep the equation balanced by doing the same operation to both sides.

By doing this, the variable \( x \) stands alone on one side of the equation, while the other side holds the solution once simplified.

Linear Equations with Constants

Linear equations often include constants, which are numbers without variables. In the given equation, \( c \) and \( m \) are constants. Dealing with linear equations with constants requires you to separate these constants from the variables during the solving process.

Follow these steps to manage constants:

• Move each constant to one side of the equation to consolidate them.
• Combine all constants together by addition or subtraction.
• After combining like terms and isolating the variable, the constants will form the number that \( x \) equals once you solve the equation.

Constants are pivotal numbers in your equation; they anchor the variables and ultimately help determine the solution.