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Chapter 2: Problem 42

Solve each equation. $$4-\frac{3}{5} x=-6$$

Short Answer

Expert verified
x = \frac{50}{3}

Step by step solution

01

- Isolate the linear term

Subtract 4 from both sides of the equation to isolate the term with the variable. \[ 4 - \frac{3}{5}x - 4 = -6 - 4 \]Simplifies to: \[ -\frac{3}{5}x = -10 \]
02

- Solve for the variable

Multiply both sides of the equation by the reciprocal of \(-\frac{3}{5}\), which is \(-\frac{5}{3}\), to solve for \(x\). \[ x = -10 \times -\frac{5}{3} \]Simplifies to: \[ x = \frac{50}{3} \]
03

- Simplify the solution

Write the final simplified solution for \(x\). \[ x = \frac{50}{3} \]

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Key Concepts

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

Isolating the Variable
Isolating the variable is an essential step in solving linear equations. This process involves manipulating the equation to get the variable on one side and constants on the other.
For the given equation: \[ 4 - \frac{3}{5}x = -6 \]
Subtract 4 from both sides to isolate the term with \(x\): \[ 4 - \frac{3}{5}x - 4 = -6 - 4 \]Simplifies to: \[ -\frac{3}{5}x = -10 \]
This step is crucial because it sets the stage for you to solve for the variable easily. Remember, you always do the same operation on both sides of the equation to maintain balance.
Linear Equations
Linear equations are algebraic expressions where the highest power of the variable is one. They are represented in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
In our exercise, the equation \[ 4 - \frac{3}{5}x = -6 \] is a linear equation because \(x\) has a power of one.
Understanding this basic form helps in recognizing and applying appropriate methods to solve it. Linear equations have straightforward solutions and are foundational in algebra.
Reciprocal Multiplication
Reciprocal multiplication involves multiplying by the reciprocal to eliminate a fraction.
After isolating the term with \(x\), we have: \[ -\frac{3}{5}x = -10 \]
To solve for \(x\), multiply both sides by the reciprocal of \(-\frac{3}{5}\), which is \(-\frac{5}{3}\): \[ x = -10 \times -\frac{5}{3} \]Simplifies to: \[ x = \frac{50}{3} \]
This operation leaves the variable by itself, making it easy to find its value.
Simplifying Fractions
Simplifying fractions is a key step in obtaining a final, clean answer. It ensures that the solution is in its simplest form. In our exercise:
After multiplying by the reciprocal, we get: \[ x = \frac{50}{3} \]
This fraction is already in its simplest form as \(\frac{50}{3}\) does not reduce further. Remember to always check if the numerator and denominator have a common factor for simplification.
Simplifying fractions makes the solution more interpretable and neat, crucial for communicating your results clearly.

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Most popular questions from this chapter

Solve each problem. Harold's mother, who lives across the street, is pouring a concrete driveway, 12 feet wide and 4 inches thick, from the street straight to her house. This is too much work for Harold to do in one day, so his mother has agreed to buy 4 cubic yards of concrete each Saturday for three consecutive Saturdays. How far is it from the street to her house?

Show a complete solution to each problem. Tis the seasoning. Cheryl's Famous Pumpkin Pie Seasoning consists of a blend of cinnamon, nutmeg, and cloves. When Cheryl mixes up a batch, she uses 200 ounces of cinnamon, 100 ounces of nutmeg, and 100 ounces of cloves. If cinnamon sells for \(\$ 1.80\) per ounce, nutmeg sells for \(\$ 1.60\) per ounce, and cloves sell for \(\$ 1.40\) per ounce, what should be the price per ounce of the mixture?

If 3 is added to every number in \((4, \infty),\) the resulting set is \((7, \infty) .\) In each of the following cases, write the resulting set of numbers in interval notation. Explain your results. a) The number \(-6\) is subtracted from every number in \([2, \infty)\) b) Every number in \((-\infty,-3)\) is multiplied by 2 c) Every number in \((8, \infty)\) is divided by 4 d) Every number in \((6, \infty)\) is multiplied by \(-2\) e) Every number in \((-\infty,-10)\) is divided by \(-5\)

Solve each problem. During World War II the Allies captured some German tanks on which the smallest serial number was \(S\) and the biggest was \(B\). Assuming the entire production of tanks was numbered 1 through \(N\), the Allies used the function \(N=B+S-1\) to estimate the number of tanks in the German army. a) Find \(N\) if \(B=2003\) and \(S=455\) b) If this formula was used to estimate \(N=1452\) and the largest serial number was \(1033,\) what was the smallest serial number?

Solve problem by using an inequality. Sewing machines. Charles wants to buy a sewing machine in a city with a \(10 \%\) sales tax. He has at most \(\$ 700\) to spend. In what price range should he look?
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