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Chapter 2: Problem 63

Solve equation, and check your solution \(7 m+6 m-4 m=63\)

Short Answer

Expert verified
The solution is \(m = 7\).

Step by step solution

01

Combine Like Terms

To simplify the left side of the equation, combine the like terms: \[7m + 6m - 4m\]
02

Simplify the Equation

Add and subtract the coefficients of the terms involving \(m\): \[ (7 + 6 - 4)m = 9m \] Now the equation is: \[9m = 63\]
03

Solve for \(m\)

To isolate \(m\), divide both sides of the equation by 9: \[ m = \frac{63}{9} \]Simplify the fraction: m = 7
04

Check Your Solution

Substitute \(m = 7\) back into the original equation to verify: \[ 7(7) + 6(7) - 4(7) = 63 \] Simplify: \[49 + 42 - 28 = 63 \] \[63 = 63 \]The solution checks out.

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Key Concepts

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

combine like terms
When solving linear equations, it's crucial to first gather all similar terms. Like terms are terms that include the same variable raised to the same power. In the equation provided, we have the terms \(7m\), \(6m\), and \(4m\). These all include the same variable \(m\). To combine them, simply add or subtract their coefficients. The coefficients here are 7, 6, and -4. Adding these together, we get \[ 7 + 6 - 4 = 9 \] Thus, \[7m + 6m - 4m \text{simplifies to} 9m\text{.}\] Combining like terms makes our equation easier to solve.
simplify equations
After combining like terms, the next goal is to simplify the equation further. In our example, once we combine \(7m\), \(6m\), and \(4m\), we get \(9m\). Therefore, the equation \[ 7m + 6m - 4m = 63 \] simplifies to \[ 9m = 63\text{.} \] Simplifying equations means to reduce them to their simplest form, often by combining like terms, as we did above. This step makes solving for the variable straightforward.
solve for variable
To solve linear equations like the one in our example, we want to isolate the variable. Our simplified equation is \[ 9m = 63\text{.} \] Here, \(m\) is multiplied by 9. To isolate \(m\), we divide both sides by 9: \[ m = \frac{63}{9}\text{.} \] Simplify the fraction to find the value of \(m\): \[ m = 7\text{.} \] By isolating the variable, finding its value becomes straightforward and easy.
check solution
To ensure your solution is correct, always substitute the value of the variable back into the original equation. For our example, we found \(m = 7\). Plugging this back in, we get \[ 7(7) + 6(7) - 4(7)\text{.} \] Simplify the terms: \[ 49 + 42 - 28 = 63 \text{.} \] Finally, verify the statement is true: \[ 63 = 63 \text{.} \] Since both sides of the equation match, our solution \(m = 7\) is indeed correct. Checking your solution helps confirm accuracy and ensures no steps were missed.

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