Problem 39 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ -3 m+2-8 m-4=-14 m+m-4 $$

Step-by-Step Solution

Answer

The solution is \(m = -1\).

Step 1: Simplify Both Sides

Start by simplifying both sides of the equation. Combine the like terms on each side. On the left side, combine \(-3m\) and \(-8m\), and combine the constants \(+2\) and \(-4\). On the right side, combine \(-14m\) and \(+m\). The equation simplifies to: \[-11m - 2 = -13m - 4\]

Step 2: Move Variables to One Side

To get all the terms with \(m\) on one side, add \(13m\) to both sides of the equation. This gives you: \[2m - 2 = -4\]

Step 3: Isolate the Variable Term

Now, isolate the term with the variable by adding \(2\) to both sides. This results in: \[2m = -2\]

Step 4: Solve for the Variable

Finally, divide both sides by \(2\) to solve for \(m\): \[m = -1\]

Step 5: Check the Solution

Substitute \(m = -1\) back into the original equation to verify the solution:Left side: \(-3(-1)+2-8(-1)-4 = 3 + 2 + 8 - 4 = 9\)Right side:\(-14(-1) + (-1) - 4 = 14 - 1 - 4 = 9\)Both sides equal \(9\), confirming the solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable, which makes the equation true. For example, consider our equation: \[-3m + 2 - 8m - 4 = -14m + m - 4\]The key to solving such an equation is to simplify and rearrange terms to isolate the variable term. The first step is to simplify both sides by combining like terms:

• On the left side, combine \(-3m\) and \(-8m\), resulting in \(-11m\); combine the constants \(+2\) and \(-4\) to get \(-2\).
• On the right side, simplify \(-14m\) and \(+m\) into \(-13m\).

This process simplifies the equation to:\[-11m - 2 = -13m - 4\] Solving equations is like untangling a knot: you systematically break it down into simpler pieces until you find what's hidden beneath.

Variable Isolation

Variable isolation involves rearranging the equation to get the variable alone on one side. In our example, the simplified equation is: \[-11m - 2 = -13m - 4\]To isolate \(m\), you must move all \(m\) terms to one side.

• Add \(13m\) to both sides to get \(2m\) on the left side:
• Now the equation is: \(-11m + 13m - 2 = -4\), which simplifies to \(2m - 2 = -4\).

The next step is to eliminate other terms. Add \(+2\) to both sides to focus solely on \(2m\):\[2m = -2\]Finally, divide by \(2\) to get:\[m = -1\]Isolating the variable is like cleaning a cluttered desk: you remove each unnecessary item so you can find what you're looking for.

Checking Solutions

Checking solutions ensures that your answer satisfies the original equation. Let’s verify that \(m = -1\) is correct by substituting it back into the original equation:Left Side:Substitute \(m = -1\) into the left part \(-3(-1) + 2 - 8(-1) - 4\):

• Calculate each term:
  • \(-3(-1) = 3\)
  • \(2\)
  • \(-8(-1) = 8\)
  • \(-4\)
• Combine: \(3 + 2 + 8 - 4 = 9\)

Right Side:Substitute into the right part \(-14(-1) + (-1) - 4\):

• Calculate:
  • \(-14(-1) = 14\)
  • \(-1\)
  • \(-4\)
• Combine: \(14 - 1 - 4 = 9\)

Both sides equal \(9\), confirming that \(m = -1\) is indeed the correct solution.Checking your work is like proofreading a paper: it's the final step to guarantee everything is right.