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Magazine Chapter 3: Problem 41
Solve each equation. $$\frac{5}{6} x+\frac{1}{4}=-\frac{9}{4}$$
Short Answer
Expert verified
The solution is \( x = -3 \).
Step by step solution
01
Isolate the Variable Term
To solve the equation \( \frac{5}{6} x + \frac{1}{4} = -\frac{9}{4} \), we first need to isolate the term containing the variable \( x \). We do this by subtracting \( \frac{1}{4} \) from both sides of the equation:\[\frac{5}{6} x + \frac{1}{4} - \frac{1}{4} = -\frac{9}{4} - \frac{1}{4}\]Simplify the equation:\[\frac{5}{6} x = -\frac{9}{4} - \frac{1}{4}\]
02
Combine Like Terms
Now, we need to combine the constants on the right side of the equation:\[-\frac{9}{4} - \frac{1}{4} = -\frac{9 + 1}{4} = -\frac{10}{4}\]Simplify the fraction:\[-\frac{10}{4} = -\frac{5}{2}\]So the equation becomes:\[\frac{5}{6} x = -\frac{5}{2}\]
03
Solve for x
To find the value of \( x \), we need to eliminate the fraction \( \frac{5}{6} \) multiplying \( x \). Multiply both sides of the equation by the reciprocal of \( \frac{5}{6} \), which is \( \frac{6}{5} \):\[\frac{6}{5} \cdot \frac{5}{6} x = -\frac{5}{2} \cdot \frac{6}{5}\]On the left side, the fractions cancel, leaving:\[x = -\frac{5 \cdot 6}{2 \cdot 5}\]Simplify the expression:\[x = -\frac{6}{2} = -3\]
04
Verify the Solution
Plug \( x = -3 \) back into the original equation to verify:\[\frac{5}{6}(-3) + \frac{1}{4} = -\frac{9}{4}\]Calculate \( \frac{5}{6}(-3) \):\[\frac{5}{6}(-3) = -\frac{15}{6} = -\frac{5}{2}\]Combine \( -\frac{5}{2} + \frac{1}{4} \):Convert \( -\frac{5}{2} \) to \( -\frac{10}{4} \) to ease addition:\[-\frac{10}{4} + \frac{1}{4} = -\frac{9}{4}\]The left side equals the right side, so \( x = -3 \) is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Isolation of Variables
In solving linear equations, isolating the variable is a key method where you aim to get the variable, like \( x \), on one side of the equation by itself. This helps in determining its exact value. Here鈥檚 the general approach:
- Identify which term includes the variable.
- Perform the inverse operation to remove any other terms from that side. For example, if a term is added to \( x \), you subtract it from both sides.
- Focus on maintaining balance in the equation by doing the same operation on both sides. This keeps the equation valid.
Fractions
Fractions often pop up in equations, and they require a little extra care. Here are a few tips to handle them effectively:
- Always ensure that you perform operations on both the numerator and the denominator when simplifying fractions.
- A common strategy is to find the least common multiple (LCM) when you're combining or comparing fractions. In this case, it helps to simplify the problem by giving fractions a common denominator.
- It鈥檚 useful to practice converting between improper fractions and mixed numbers, but improper fractions are often simpler to work with when doing algebra.
Reciprocal Multiplication
Reciprocal multiplication is a technique used to eliminate fractions when a variable is embedded within them. It鈥檚 an essential skill to clear fractions from the equation, making it simpler to find the solution.
- The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). When you multiply a fraction by its reciprocal, the result is 1 because \( \frac{a}{b} \times \frac{b}{a} = 1 \).
- By multiplying both sides of the equation by the reciprocal, you effectively "remove" the fraction from the side with the variable.
- This process simplifies the equation and should lead directly to solving for the variable.
