Problem 74 - Algebra 2

Question

Solve each equation and check the solution. $$ 2(m-3)=-4 $$

Step-by-Step Solution

Answer

The solution is m = 1.

Step 1: Expand the Equation

Distribute the 2 to both terms inside the parenthesis: 2(m-3) becomes 2*m - 2*3, which simplifies to 2m - 6.

Step 2: Isolate the Variable

To isolate m, add 6 to both sides of the equation: 2m - 6 + 6 = -4 + 6. This simplifies to 2m = 2.

Step 3: Divide by Coefficient of the Variable

Divide both sides of the equation by 2 to solve for m: 2m / 2 = 2 / 2. This gives us m = 1.

Step 4: Check the Solution

Substitute 1 for m in the original equation to verify the solution: 2(1 - 3) = 2(1) - 2(3) = 2 - 6 = -4, which confirms that m = 1 is correct.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is vital in solving linear equations. An algebraic expression is a combination of numbers, variables (like m), and arithmetic operations (addition, subtraction, multiplication, and division). In the example exercise, 2(m-3) is an algebraic expression which contains a multiplication operation implied between the number 2 and the parenthesis, and a subtraction within the parenthesis. Simplifying algebraic expressions is the first step in solving equations, wherein you apply the distributive property—here, multiplying 2 with each term inside the parenthesis, leading to 2m - 6.

Storytelling, especially with word problems, can also help students relate and better understand abstract concepts. For instance, imagine you have 2 bags containing m marbles each, but you must also take out 6 marbles, this forms the understanding of combining like terms and leads to the expression 2m - 6.

Isolate the Variable

Isolating the variable is akin to finding a treasure by eliminating all distractions or false trails. The goal is to have the variable on one side of the equation and the constants on the other. In our exercise, to isolate m, we need to remove the -6 by doing the inverse operation, which is adding 6. This is outlined as 2m - 6 + 6 = -4 + 6. Once the -6 has been neutralized or 'canceled out', we are left with 2m = 2. Now, m is not alone yet, as it is being multiplied by 2. Performing the opposite operation, which is division in this case, results in the variable standing alone: m = 1.

Illustrating this concept with everyday analogies helps, such as comparing it to balancing a seesaw. Each action you perform on one side, you must equally do on the other to keep it balanced, ensuring the equation remains true.

Equation Solving Steps

Solving an equation is a process that involves several studies and procedural steps. To ensure a proper understanding, let's unpack the steps with our example equation: 2(m-3)=-4. The first step is to expand the equation using the distributive property, resulting in 2m - 6. Next, we aim to isolate the variable, which involves adding 6 to both sides. Following this, we divide by the coefficient of the variable—2 in this case—to find m = 1. The final and critical step is to check the solution by substituting the variable back into the original equation to ensure the equality holds. If m = 1 is the correct solution, then the left-hand side should equal the right-hand side when we perform the substitution, which verifies our solution.

Remember, inclusivity in explanation styles by using diagrams, verbal explanations, and written methods cater to different learning styles and allow for a deeper understanding of equation solving steps.