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Chapter 2: Problem 44

Solve each equation. $$2(8 m+3)-15 m-4=9(m+6)-2(m-1)-7 m$$

Short Answer

Expert verified
m = 54

Step by step solution

01

Distribute and Combine Like Terms

Distribute the numbers outside the parentheses and then combine the like terms on both sides of the equation. The equation becomes: 2(8m + 3) - 15m - 4 = 9(m + 6) - 2(m - 1) - 7mThis simplifies to:16m + 6 - 15m - 4 = 9m + 54 - 2m + 2 - 7m
02

Simplify Both Sides

Combine like terms on both sides to further simplify the equation. This results in:16m - 15m + 6 - 4 = 9m - 2m - 7m + 54 + 2Which simplifies to:m + 2 = 0m + 56
03

Isolate the Variable

To isolate m, subtract 2 from both sides of the equation:m + 2 - 2 = 56 - 2This results in:m = 54

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Key Concepts

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

Distributive Property
The distributive property is a key step when solving linear equations, especially when parentheses are involved.
It allows you to 'distribute' a number outside the parenthesis to each term inside. For example, in the equation 2(8m + 3), use the distributive property to multiply 2 by both 8m and 3.
This results in 16m + 6. Similarly, for 9(m + 6), apply the distributive property to get 9m + 54, and for -2(m - 1), you distribute -2 to get -2m + 2.
Always remember to distribute both positive and negative signs with equal care.
Combining Like Terms
Once you've used the distributive property, the next step is to combine like terms.
Like terms are terms that have the same variable raised to the same power or just the constants. For example, in the simplified equation 16m + 6 - 15m - 4, combine the terms with 'm' and the constant terms separately.
This results in 1m + 2. Similarly, on the other side, combining 9m - 2m - 7m gives you 0m, and combining 54 + 2 gives you 56.

Combining like terms simplifies the equation and makes it easier to isolate variables.
Always align like terms vertically to avoid mistakes.
Isolating Variables
Isolating the variable is the final step to solve the equation. In our example, we had simplified the equation to m + 2 = 56.
To isolate 'm', you need to get rid of the constant on the same side as the variable. You can do that by subtracting 2 from both sides of the equation. This gives m + 2 - 2 = 56 - 2, which simplifies to m = 54.

Isolating the variable often involves adding or subtracting terms on both sides of the equation.
Remember to always perform the same operation on both sides to maintain the balance of the equation.
  • Subtract or add to remove constants.
  • Use multiplication or division to deal with coefficients of variables.

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