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Chapter 3: Problem 32

Solve each equation by first clearing fractions or decimals. $$0.2(12)+0.08 z=0.12(z+12)$$

Short Answer

Expert verified
We can eliminate decimals from the given equation by multiplying by 100: \[100 \times (0.2(12) + 0.08z) = 100 \times (0.12(z + 12))\]. This results in \[20(12) + 8z = 12(z + 12)\]. Expanding and simplifying, we get \[240 + 8z = 12z + 144\]. Moving terms containing z to one side, we obtain \[240 = 4z + 144\]. Isolating the variable z gives \[96 = 4z\]. Finally, solving for z yields \(z = 24\).

Step by step solution

01

Eliminate decimals

In the given equation, all decimals have multiples of 100. Therefore, to eliminate decimals, multiply both sides of the equation by 100: \[100 \times (0.2(12) + 0.08z) = 100 \times (0.12(z + 12))\]
02

Simplify the equation

Multiply both sides of the equation by 100 and simplify: \[20(12) + 8z = 12(z + 12)\]
03

Expand the equation

Expand both sides of the equation: \[240 + 8z = 12z + 144\]
04

Move all terms containing z to one side

Subtract 8z from both sides of the equation: \[240 = 4z + 144\]
05

Isolate the variable z

Subtract 144 from both sides of the equation: \[96 = 4z\]
06

Solve for z

Divide both sides of the equation by 4: \[z = 24\] The solution to the equation is \(z = 24\).

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Key Concepts

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

Solving Equations
To solve linear equations, we follow a systematic approach to find the unknown variable. Linear equations are mathematical expressions that equate two expressions, typically involving a single variable raised to the power of one. Solving them requires balance; anything done to one side of the equation must be done to the other.

Here, the equation can involve operations like addition, subtraction, multiplication, and division. The goal is to simplify the equation progressively to isolate the variable, making it the subject of the formula.
Clearing Decimals
Decimals in equations can make calculations tricky, so it's often easier to clear them. We do this by multiplying the entire equation by a power of 10 (like 10, 100, or 1000), depending on the decimal places in the terms.

This process transforms the decimals into whole numbers, simplifying the process of solving the equation. For example, in the equation \(0.2(12) + 0.08 z = 0.12(z + 12)\), multiplying by 100 eliminates all decimals: \(20(12) + 8z = 12(z + 12)\).

It鈥檚 crucial to apply this multiplication evenly across all terms to maintain the equation鈥檚 balance.
Step by Step Solution
A systematic, step-by-step solution method is essential for solving equations accurately and efficiently. It involves:
  • Identifying operations involved: multiplication, addition, etc.
  • Clearing fractions or decimals if present.
  • Simplifying each side of the equation by expanding and combining like terms.
  • Rearranging the equation to move all instances of the variables to one side, and constants to the other.
  • Using inverse operations to solve for the variable.
With each step clearly identified, students can track their work and understand each transformation in the equation. This approach also helps in identifying and correcting mistakes.
Isolate Variable
Isolating the variable is the key target in solving any linear equation. It involves manipulating the equation to get the unknown variable on one side, and all numbers on the opposite side.

To isolate the variable, we perform operations that "undo" others. For instance, if multiplication is involved on the variable, division will be used. In our exercise, after clearing decimals and simplifying, the steps involved:
  • Subtracting 8z from both sides, to gather all z terms on one side.
  • Subtracting constants from the side with the variable, to leave z alone.
  • Finally, dividing by the coefficient of z to solve, giving \(z = 24\).
This clear manipulation ensures that you uncover the value of the unknown variable efficiently.

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Most popular questions from this chapter

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) Twice a number decreased by seven is \(-13 .\) Find the number.

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