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Chapter 5: Problem 34

Solve the equations. $$-1.5 m+14.26=0.2(18-m)$$

Short Answer

Expert verified
The value of \(m\) is 8.2.

Step by step solution

01

Distribute the 0.2

First, distribute the 0.2 to both terms within the parentheses on the right side of the equation. This gives us: \(-1.5m +14.26 = 0.2(18) - 0.2(m)\) Simplifying the distributed terms: \(-1.5m + 14.26 = 3.6 - 0.2m\)
02

Combine like terms

Next, we want to combine the terms involving \(m\) on one side of the equation: Add \(0.2m\) to both sides: \(-1.5m + 0.2m + 14.26 = 3.6\) Combine the \(m\) terms: \(-1.3m + 14.26 = 3.6\)
03

Isolate the variable term

Subtract 14.26 from both sides to isolate the term with \(m\): \(-1.3m = 3.6 - 14.26\) Simplify the right side: \(-1.3m = -10.66\)
04

Solve for m

Finally, divide both sides by -1.3 to solve for \(m\): \(m = \frac{-10.66}{-1.3}\) Simplify the fraction: m = 8.2

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Key Concepts

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

distributive property
When you face an equation where a number is multiplied by a parenthesis, you need to apply the distributive property. This means you multiply the number by each term inside the parenthesis separately. In our example:
  • The number 0.2 is distributed to both 18 and \( -m \).
  • This gives us \(-1.5m + 14.26 = 0.2(18) - 0.2(m)\).
  • By multiplying the terms, it simplifies to \(-1.5m + 14.26 = 3.6 - 0.2m\).
Using the distributive property helps in breaking down and simplifying complex equations.
combining like terms
Once you've applied the distributive property, the next step is to simplify the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our example:
  • We need to combine the terms with \(m\) together.
  • We add \(0.2m\) to both sides to start getting all variable terms together: \(-1.5m + 0.2m + 14.26 = 3.6\).
  • This combines to \(-1.3m + 14.26 = 3.6\).
Combining like terms makes the equation much easier to solve.
isolating the variable
The key to solving any equation is to isolate the variable you're trying to solve for. In this case, we are solving for \(m\). To isolate \(m\):
  • Subtract 14.26 from both sides: \(-1.3m = 3.6 - 14.26\).
  • Simplify the right side: \(-1.3m = -10.66\).
Now, \(m\) is isolated on one side, ready to be solved.
simplifying fractions
The final step often involves simplifying a fraction to find the value of the variable. To find \(m\):
  • Divide both sides by \(-1.3\): \m = \frac{-10.66}{-1.3}\.
  • Simplify the fraction: \m = 8.2\.
Simplifying fractions ensures the solution is in its simplest and most understandable form.

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