Problem 7 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ 19 m=-19 $$

Step-by-Step Solution

Answer

The solution to the equation \(19m = -19\) is \(m = -1\).

Step 1: Isolate the Variable

Start by isolating the variable 'm' on one side of the equation. To do this, divide both sides of the equation by 19. So, it gives us the equation \(m = -19 / 19\).

Step 2: Solve for m

Now solve the equation for the variable 'm'. \(-19 / 19\) simplifies to -1. So, \(m = -1\).

Step 3: Check the Solution

Substitute \(m = -1\) into the original equation \(19m = -19\) to check. Substituting it in gives \(19*(-1) = -19\), which simplifies to \(-19 = -19\). The left side of the equation equals the right side of the equation, confirming that \(m = -1\) is the correct solution.

Key Concepts

Isolate the Variable

When solving linear equations, your primary goal is to isolate the variable. This means you want to get the variable alone on one side of the equation. By doing this, you can determine its value by performing mathematical operations step by step. For the equation \(19m = -19\), you need the variable 'm' by itself. This can be achieved by dividing both sides of the equation by 19. Remember, whatever action you do on one side of the equation, you must do on the other to maintain balance.

• Divide both sides by 19.
• This helps you transform the equation to \(m = -19/19\).
• Resulting in \(m = -1\) after simplification.

By isolating 'm', you now have the solution where 'm' equals -1. This step is crucial as it simplifies the equation directly to the variable we are solving for.

Substitution Method

The substitution method allows us to verify our solution. Once you have found the value of the variable, plug it back into the original equation. This helps confirm that the solution is consistent and correct. For our exercise, we've discovered that \(m = -1\). Let's ensure this value satisfies the initial equation:

• Original Equation: \(19m = -19\)
• Substitute \(m = -1\) into the equation.
• Calculation: \(19 \times (-1) = -19\)

Substituting back confirms that the equation holds true because both sides equal the same value \(-19\). The substitution method serves as a practical tool to double-check your results.

Verify Solutions

Verifying your solutions is the final and crucial step when solving equations. It ensures the correctness of your solution, instilling confidence in your mathematical processes. To verify, you need to check if the calculated result translates into a true statement when substituted back into the original equation:

• Plug the solution, \(m = -1\), back into \(19m = -19\).
• Solve to confirm both sides of the equation are equal.
• Reflects as \(-19 = -19\), showing they are indeed balanced.

Verification not only confirms accuracy but also helps identify manual errors early on. By thoroughly checking your solutions, you are more likely to avoid mistakes and solidify your understanding of solving linear equations.