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Chapter 2: Problem 72

Solve each equation, and check your solution. $$ 9(2 m-3)-4(5+3 m)-5(4+m)=-3 $$

Short Answer

Expert verified
The solution is \( m = 64 \).

Step by step solution

01

- Distribute the Constants

Apply the distributive property to each term to remove the parentheses: \[ 9(2m - 3) - 4(5 + 3m) - 5(4 + m) = -3 \] becomes\[ 18m - 27 - 20 - 12m - 20 - 5m = -3 \]
02

- Combine Like Terms

Combine like terms on the left side of the equation: \[ 18m - 12m - 5m - 27 - 20 - 20 = -3 \] Simplifying, we get \[ m - 67 = -3 \]
03

- Solve for m

Solve for the variable by isolating m: \[ m - 67 = -3 \] Add 67 to both sides: \[ m = 64 \]
04

- Check the Solution

Substitute \( m = 64 \) back into the original equation to check if it satisfies the equation: \[ 9(2(64) - 3) - 4(5 + 3(64)) - 5(4 + 64) \ = 9(128 - 3) - 4(5 + 192) - 5(68) = 9(125) - 4(197) - 5(68) \ = 1125 - 788 - 340 = -3 \] Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

distributive property
One of the first steps in solving linear equations is to simplify the terms inside parentheses. The distributive property helps us do this. It's a way to rewrite expressions by distributing a factor over terms inside the parentheses. For example, in our equation, we have terms such as 9(2m - 3). Using the distributive property, this becomes 18m - 27. This is because we multiply 9 by each term inside the parentheses. Similarly, we apply the distributive property to -4(5 + 3m). This results in -20 - 12m. And, for -5(4 + m), we get -20 - 5m. Applying the distributive property is crucial for reducing the equation and making it easier to solve.
combining like terms
After using 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. For example, in the expression 18m - 27 - 20 - 12m - 20 - 5m, we need to group together all the m terms. This allows us to simplify the equation. First, combine the m terms: 18m - 12m - 5m. This simplifies to m. Second, combine the constant terms: -27 - 20 - 20. This simplifies to -67. Now our equation looks like this: m - 67. Combining like terms helps in organizing the equation properly for further steps.
isolating variables
The final step in solving a linear equation is to isolate the variable. This means getting the variable by itself on one side of the equation. In our simplified equation, m - 67 = -3, we need to isolate m. To do this, we add 67 to both sides of the equation: m - 67 + 67 = -3 + 67. This gives us: m = 64. Once the variable is isolated, we can state the solution with certainty. To ensure our solution is correct, we substitute m = 64 back into the original equation and check if both sides are equal. This confirms that our solution is indeed correct. Isolating the variable is a key step in solving any linear equation.

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