Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{10 x+3}{5 x+6}=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution to the equation is x = 0.

Step 1: Cross Multiplication

Start by cross multiplying: \((10x+3)*(2) = (5x+6)*1. Simplify this to get: 20x+6 = 5x+6.

Step 2: Simplify

Next, combine like terms and isolate x: 20x - 5x = 6 - 6. This simplifies to: 15x = 0.

Step 3: Solve for x

To solve for x, divide both sides of the equation by 15: x = 0/15 which simplifies to x = 0.

Step 4: Check solution

To check if the solution is correct, plug x=0 back into original equation: \(\frac{10*0+3}{5*0+6} = \frac{1}{2}\). This simplifies to \(\frac{3}{6} = \frac{1}{2}\), which is a valid equation, so x = 0 is the solution.

Key Concepts

Cross Multiplication

When faced with an equation involving fractions, such as \( \frac{10x+3}{5x+6}=\frac{1}{2} \), a useful technique to eliminate the fractions is cross multiplication. This method allows us to work with a simpler, whole number equation. Here's how you do it:

• Take the numerator of the first fraction (\(10x+3\)) and multiply it by the denominator of the second fraction (2).
• Then, take the numerator of the second fraction (1) and multiply it by the denominator of the first fraction (\(5x+6\)).

This gives us the equation:\[(10x+3) \times 2 = 1 \times (5x+6)\]By carrying out these multiplications, we obtain:\[20x + 6 = 5x + 6\]Cross multiplication transforms our original fraction-based equation into an easier form to solve, making the path clear to finding the solution.

Simplifying Equations

Once you've used cross multiplication, your task is to simplify the resulting equation. Simplifying means reducing the equation step by step until you isolate the variable. Let's walk through the process using our equation:\[20x + 6 = 5x + 6\]Here's how you simplify it:

• First, move all terms containing \(x\) to one side of the equation by subtracting \(5x\) from both sides, which gives you \(20x - 5x + 6 = 6\).
• This simplifies further to \(15x + 6 = 6\).
• Next, eliminate the constant on the left side by subtracting 6 from both sides, resulting in \(15x = 0\).

At this point, the equation is simplified, and the variable \(x\) is isolated, ready for the final step of solving.

Checking Solutions

Checking your solution is a crucial step in solving equations to ensure your answer is correct. It's like double-checking your work. Once you have found a solution, substitute it back into the original equation to see if it holds true.In our case, the solution we found was \(x = 0\). Substituting \(x = 0\) back into the original equation, \( \frac{10x+3}{5x+6} = \frac{1}{2} \), goes like this:

• Replace \(x\) with 0: \(\frac{10(0)+3}{5(0)+6} = \frac{3}{6}\).
• Simplify the fraction: \(\frac{3}{6}\) reduces to \(\frac{1}{2}\).

The left side of the equation equals the right side, confirming \(\frac{1}{2} = \frac{1}{2}\). Thus, our solution \(x = 0\) is correct, and no steps were skipped or miscalculated.