Problem 32 - Algebra 1

Question

Solve the equation. $$\frac{-3}{x+7}=\frac{2}{x+2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{20}{5}\)

Step 1: Finding the common denominator

To eliminate fractions normalize both sides of the equation to a common denominator. The common denominator for \(x + 7\) and \(x + 2\) can be found by multiplying the two denominators together, giving \(x + 7\) and \(x + 2\). Thus, the equation becomes:\[-3(x + 2) = 2(x + 7)\]

Step 2: Solving the equation

Expand and simplify the equation:\[-3x - 6 = 2x + 14\]Solve for \(x\) by adding \(3x\) to both sides of the equation and subtracting \(14\) from both sides of the equation we would get:\[-3x + 3x - 14 - 6 = 2x + 3x + 14 - 14\]Simplify to get final answer:\[x = -\frac{20}{5}\]

Step 3: Checking the solution

To verify if the solution \(x = -\frac{20}{5}\) is correct, substitute it into the original equation and simplify:\[\frac{-3}{-\frac{20}{5}+7} = \frac{2}{-\frac{20}{5}+2}\]Simplify both sides. If the equation holds true, then the solution \(x = -\frac{20}{5}\) is correct.

Key Concepts

Common Denominator

When you are solving rational equations, finding a common denominator is key. Think of the denominator as the basis on which you will unify both sides of your equation. In our example, we have two fractions \,\( \frac{-3}{x+7} \,\) and \,\( \frac{2}{x+2} \,\).To find the common denominator, multiply these two existing denominators together, giving you \,\((x + 7) \times (x + 2)\).

• This allows you to eliminate fractions by multiplying both sides of the equation by this common denominator.
• So, \,\(-3(x + 2) = 2(x + 7)\). Now, you can work with a simpler equation without fractions.

Remember that dealing with denominators can be tricky if you overlook them. Always double-check to ensure your common denominator is correct.

Expand and Simplify

Once you've found a common denominator and eliminated the fractions, the next step is to expand and simplify the equation. Look at the expression \,\(-3(x + 2) = 2(x + 7)\). Distribute the multiplication across both terms inside the parentheses.

• For the left side: \,\(-3x - 6\).
• For the right side: \,\(2x + 14\).

Once expanded, gather all variables on one side of the equation and constants on the other. In this example, add \,\(3x\) to both sides and subtract \,\(14\) from both sides:

• \,\(-3x + 3x - 6 - 14 = 2x + 3x\)

After simplifying all like terms, you'll find that \,\(x = -\frac{20}{5}\). Simplifying the constants and coefficients until you isolate the variable will always lead you to your solution.

Check the Solution

After calculating the solution, it is crucial to check if it is correct. Substitution back into the original equation helps verify the validity of your solution. Our solution is \,\(x = -\frac{20}{5}\), substitute this value into the original rational equation:

• For the left side: \,\(\frac{-3}{-\frac{20}{5} + 7}\).
• For the right side: \,\(\frac{2}{-\frac{20}{5} + 2}\).

Simplify each expression to check if both sides are equal. If they match, you have confirmed your solution is correct. Always remember, replacing and verifying should reflect back to your original equation; if both sides are equal, your solution is valid!Checking ensures that mistakes were caught and confirms your proficiency in solving rational equations.