Question

Solve the equation if possible. $$ -7+4 m=6 m-5 $$

Short answer

The solution to the equation -7+4m=6m-5 is \( m = -1 \).

Step-by-step solution

Simplify each side of the equation

The first step is to simplify the equation by collecting like terms, i.e terms with \( m \), on one side of the equation, and terms without \( m \) on the other side. Thus, add \( -4m \) to both sides to neutralize \( 4m \) on the left side and add \( 7 \) to both sides to neutralize \( -7 \) on the right side. This gives the equation: \( -7+4m+(-4m)+7=6m-5+(-4m)+7 \).

Begin to isolate the variable

Simplify the equation obtained in Step 1. On the left-hand-side, \( 4m \) and \( -4m \) cancel out, leaving \( -7+7 \); and on the right-hand-side, combine like terms to get \( 6m-4m \) and \( -5+7 \). This simplifies to: \( 0 = 2m +2 \).

Isolate the variable

To further isolate the variable \( m \), subtract 2 from both sides. The equation becomes: 2m= -2. Finally, to solve for \( m \), divide both sides of the gained equation by 2 to get: \( m = -1 \).

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Examples of operation symbols are additions (+), subtractions (−), multiplications (×), and divisions (÷). In the context of our equation \( -7+4m=6m-5 \) from the original exercise, \( 4m \) and \( 6m \) are algebraic expressions involving the variable \( m \) representing an unknown quantity, while \( -7 \) and \( -5 \) are constant terms.

Isolate the Variable

When we isolate the variable, we manipulate the equation to get the variable on one side and the constants on the other. The goal is to 'solve for' the variable, which means to find the value of the variable that makes the equation true. In the exercise, we follow steps to transform the equation \( -7+4m=6m-5 \) into a simpler form where \( m \) stands alone. By adding \( -4m \) to both sides, as shown in Step 1, and then adding 7 to both sides, we get \( 0 = 2m+2 \) in Step 2. Reducing the equation further in Step 3 by subtracting 2 from each side helps us to completely isolate \( m \) and find its value \( m = -1 \).

For better comprehension, imagine each step as a way to keep the scales balanced: whatever is done to one side of the equals sign must also be done to the other side, in order to maintain the balance, which is essential in solving equations.

Collecting Like Terms

The process of collecting like terms involves combining terms that have the same variables raised to the same power. In simple terms, if two or more terms have the same variable part, we can add or subtract their coefficients to simplify the expression. Looking back at the solution provided for the given equation, we added \( -4m \) to the \( 6m \) on the right side and eliminated \( 4m \) on the left side, which is part of collecting like terms. This resulted in \( 2m \) on the right side. Similarly, by adding 7 to both sides, we collected the constant terms, turning \( -7 \) and \( -5 \) into a positive 2 on the right side.

To practice this, list all like terms together and combine them into a single term. For instance, \( 3x \) and \( 5x \) would become \( 8x \) since they both contain the variable \( x \). This step is crucial in simplifying equations and making them easier to solve.