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Chapter 3: Problem 33

Solve each equation by first clearing fractions or decimals. $$0.04 s+0.03(s+200)=27$$

Short Answer

Expert verified
We can solve the equation \(0.04 s+0.03(s+200)=27\) by following these steps: 1. Clear decimals by multiplying the entire equation by 100. 2. Distribute 100 to each term inside the parentheses. 3. Distribute the 3 to the terms inside the parentheses. 4. Combine like terms. 5. Get the variable term by itself by subtracting 600 from both sides. 6. Solve for the variable by dividing by 7. After performing these steps, we find that the value of the variable \(s\) is \(300\).

Step by step solution

01

Clear decimals by multiplying the entire equation by 100

To remove the decimals from the equation, multiply the entire equation by 100: $$100(0.04 s+0.03(s+200))=100(27)$$
02

Distribute 100 to each term inside the parentheses

Multiply 100 with each term inside the parentheses on both sides of the equation: $$100(0.04 s)+100(0.03(s+200))=100(27)$$ $$4s + 3(s+200) = 2700$$
03

Distribute the 3 to the terms inside the parentheses

Distribute the 3 on the left side of the equation to the terms inside the parentheses: $$4s + 3s + 600 = 2700$$
04

Combine like terms

Combine the terms with the variable s on the left side of the equation: $$7s + 600 = 2700$$
05

Get the variable term by itself by subtracting 600 from both sides

To isolate the term with the variable, subtract 600 from both sides of the equation: $$7s + 600 - 600 = 2700 - 600$$ $$7s = 2100$$
06

Solve for the variable by dividing by 7

Divide both sides of the equation by 7 to solve for the variable s: $$\frac{7s}{7} = \frac{2100}{7}$$ $$s = 300$$ The value of the variable s is 300.

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Key Concepts

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

Solving Equations
Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations typically take the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. The goal is to isolate the variable on one side of the equation.
  • Start by simplifying each side of the equation if necessary, which can involve distributing constants and combining like terms.
  • Then, perform inverse operations to both sides to isolate the variable, such as adding, subtracting, multiplying, or dividing by a constant.
  • Always ensure that what you do to one side of the equation, you do to the other to maintain equality.
Understanding the properties of equality and operations is key. This ensures that the structure of the equation is preserved as you work towards finding the solution.
Clearing Decimals
Decimals in equations can make calculations more complex, so a common strategy is to eliminate them. The easiest way to do this is by multiplying the entire equation by a power of 10, which shifts the decimal point to the right. This essentially transforms a decimal number into a whole number, simplifying further operations.
  • Identify the highest decimal place in the equation.
  • Multiply every term by 10, 100, or 1000, depending on the decimal place, to clear the decimals.
For example, in the equation \(0.04s + 0.03(s + 200) = 27\), multiplying by 100 moves the decimals two places to the right, resulting in the simpler equation: \(4s + 3(s + 200) = 2700\). This modified equation is much easier to handle mathematically.
Distributive Property
The distributive property is a useful algebraic property that allows you to multiply a single term across terms within parentheses. This property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equivalent to \(ab + ac\). It is very helpful in breaking down expressions to make them easier to work with.
  • Apply the distributive property by multiplying the external term to each term inside the parentheses.
  • This expansion of terms helps in simplifying and solving the equation.
In our example, after clearing the decimals, the expression \(3(s + 200)\) uses the distributive property to become \(3s + 600\), making the entire equation \(4s + 3s + 600 = 2700\).
Combining Like Terms
Combining like terms is the process of simplifying expressions in an equation by merging terms that have the same variables raised to the same power. This step is vital to simplify equations and make them easier to solve.
  • Look for terms that have the same variable component; these are your like terms.
  • Add or subtract coefficients of like terms to combine them into a single term.
In our example, the terms \(4s\) and \(3s\) are like terms, as both terms have the variable \(s\). These are combined to form \(7s\), transforming the equation to \(7s + 600 = 2700\). The equation is now simpler, with fewer terms, making it easier to isolate and solve for \(s\).

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Most popular questions from this chapter

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