Problem 87 - Algebra and Trigonometry

Question

Solve the equation for the indicated variable. \(\frac{a x+b}{c x+d}=2 ;\) for \(x\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{2d - b}{a - 2c}\).

Step 1: Cross multiply the equation

Start by cross-multiplying to eliminate the fraction. This gives: \((a x + b) = 2(c x + d)\)

Step 2: Distribute and rearrange

Distribute the 2 on the right side: \(a x + b = 2c x + 2d\). Next, rearrange terms to get all terms involving \(x\) on one side: \(a x - 2c x = 2d - b\)

Step 3: Simplify the equation

Combine like terms on the left side: \((a - 2c) x = 2d - b\)

Step 4: Solve for x

Isolate \(x\) by dividing both sides by \(a - 2c\): \[x = \frac{2d - b}{a - 2c}\].

Key Concepts

Cross Multiplication

Cross multiplication is a method used to eliminate fractions from an equation. It involves multiplying each side of the equation by the denominator of the fraction on the opposite side. In our given exercise, we start with the equation \(\frac{a x+b}{c x+d}=2\). To use cross multiplication here, we multiply both sides by \((c x + d)\). This results in the equation \((a x + b) = 2(c x + d)\). By performing this step, we remove the fraction and simplify our equation, making it easier to solve. Remember these two points while cross multiplying:

• Ensure each side of the equation is multiplied by the entire denominator from the opposite side.
• Only use cross multiplication if there’s a single fraction on each side or one side equates to a fraction.

Distributive Property

The distributive property is used in mathematics to multiply a single term and two or more terms inside a set of parentheses. This property states that \(a(b + c) = ab + ac\). In our solution, we apply this to the equation \((a x + b) = 2(c x + d)\), where we need to distribute the number 2 across \(c x + d\). The equation thus becomes \(a x + b = 2c x + 2d\). Use the distributive property whenever you encounter scenarios where a term outside a parenthesis needs to be multiplied with each term inside the parenthesis. This step helps in expanding and simplifying equations for further operations, such as combining like terms.

Variable Isolation

Variable isolation is a crucial concept in algebra where the goal is to get the variable (in this case \(x\)) by itself on one side of the equation. After applying the distributive property, we have the equation \(a x + b = 2c x + 2d\). To isolate the variable \(x\), we need to gather all terms involving \(x\) on one side and constant terms on the other. By subtracting \(2c x\) from both sides, the equation becomes \(a x - 2c x = 2d - b\). Here are some tips for variable isolation:

• Always perform the same operation on both sides of the equation to maintain equality.
• Be strategic in choosing which terms to move to isolate the variable efficiently.

Variable isolation often requires careful rearrangement of terms, so practice is key!

Equation Simplification

Simplifying an equation involves combining like terms and making it as straightforward as possible. In the given exercise, after isolating potential terms for \(x\), you reach the equation \((a - 2c) x = 2d - b\). This is simplified further by recognizing \(a x\) and \(-2c x\) are like terms, allowing you to combine them as \((a - 2c)x\). The final step in simplifying is to solve for \(x\) by dividing both sides of the equation by \((a - 2c)\), which gives us \[x = \frac{2d - b}{a - 2c}\]. Key concepts in equation simplification include:

• Always combine like terms when possible (those that have the same variable raised to the same power).
• Simplify fractions where possible to find a cleaner solution.