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Chapter 4: Problem 8

Solve the equations and inequalities. $$\frac{u}{9}=-\frac{5}{12}$$

Short Answer

Expert verified
u = -\frac{15}{4}

Step by step solution

01

Set Up the Equation

Start with the given equation: divide both sides of the equation by 9: \[ \frac{u}{9} = -\frac{5}{12} \]
02

Multiply Both Sides by 9

To isolate the variable \( u \), multiply both sides of the equation by 9:\[ u = -\frac{5}{12} \times 9 \]
03

Simplify the Right Side

Simplify the expression on the right side by multiplying and reducing the fraction if necessary:\[ u = -\frac{5 \times 9}{12} \]\[ u = -\frac{45}{12} \]Divide the numerator and the denominator by their greatest common divisor, which is 3:\[ u = -\frac{15}{4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables is a fundamental technique in solving linear equations. The goal here is to get the variable by itself on one side of the equation. In the given problem, we start with the equation \(\frac{u}{9} = -\frac{5}{12} \). To isolate \( u \), we need to eliminate other numbers or fractions attached to it. This can often require using inverse operations like addition, subtraction, multiplication, or division. Here, we multiply both sides by 9 to isolate \( u \). This action keeps the equation balanced while simplifying our task to focus on solving for \( u \).
Multiplying Fractions
Multiplying fractions is an essential skill. When encountering fractions in equations, you'll often need this operation. In our example, after isolating \( u \), we multiply \(-\frac{5}{12} \) by 9. Here鈥檚 the general rule: When you multiply a fraction by a whole number, multiply the numerator (top part) of the fraction by that whole number. Then simplify if necessary. So, for our problem: \( -\frac{5}{12} \times 9 = -\frac{5 \times 9}{12} = -\frac{45}{12} \). This can be further simplified, which brings us to our next key concept.
Simplifying Fractions
Simplifying fractions makes them easier to work with and understand. Simplifying involves dividing the numerator and the denominator by their greatest common divisor (GCD). In our example, \(-\frac{45}{12} \), the GCD of 45 and 12 is 3. Dividing both the numerator and the denominator by 3 gives us: \(-\frac{45 \div 3}{12 \div 3} = -\frac{15}{4} \). Breaking down fractions into simplest terms ensures that the answer is clear and precise. Simplified fractions are easier to interpret and compare.

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Most popular questions from this chapter

Solve the equations and inequalities. $$\frac{2}{5}<\frac{2 x+3}{5}-\frac{3 x+1}{7}<\frac{6}{7}$$

Solve the equations and inequalities. $$3(y+2)+\frac{y+3}{5}=\frac{9 y+8}{2}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Takima has a semester average of \(94 .\) She has four exam scores of \(88,85,93\) and \(95 .\) If her final exam counted for 3 exams, find her score on the final.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A store advertises a \(20 \%\) sale on skirts. If the reduced price range is between \(\$ 12.60\) and \(\$ 20.76,\) what was the original range of prices?

Solve the equations and inequalities. $$\frac{t}{6}+\frac{t+3}{5}+\frac{t-2}{10}=t-6$$
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