Problem 39 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ -7+4 m=6 m-5 $$

Step-by-Step Solution

Answer

The solution to the equation is \(m = 6\). This equation has one solution.

Step 1: Simplify the equation

The given equation can be rewritten by gathering like terms together on each side: \(4m + 7 = 6m - 5\).

Step 2: Further simplify

Subtract \(4m\) from both sides to isolate \(m\) terms on one side of the equation: \(7 = 6m - 4m - 5\).

Step 3: Simplify again

Simplify the right side to get \(7 = 2m - 5\).

Step 4: Solve for m

To solve for \(m\), add 5 to both sides: \(7 + 5 = 2m\). Hence, \(m = 6\).

Key Concepts

One Solution

In mathematics, a linear equation is said to have "one solution" when it equates to a single, unique value for the variable involved. Let's consider the original exercise: -7 + 4m = 6m - 5. This equation involves a single variable, which is "m". To determine if there is one unique solution, we simplify and rearrange the equation step-by-step until "m" is isolated:

• Start by rearranging terms: 4m + 7 = 6m - 5.
• Subtract 4m from both sides to bring terms involving "m" to one side.
• This simplification leads to 7 = 2m - 5.
• Add 5 to both sides: 7 + 5 = 2m.
• Solve for "m" by dividing both sides by 2. That's when we find that m = 6.

So, the solution to this equation is that "m" equals 6, which is a single, unique value and therefore, the equation has one solution. The critical aspect of an equation having one solution is that there is exactly one specific outcome for the variable.

No Solution

An equation is said to have "no solution" when there is no possible value for the variable that can satisfy the equation. This situation typically arises when, during the simplification process, you end up with a contradiction, such as a false statement. For instance, consider an equation from a different scenario: 3x + 5 = 3x - 10. If we try to solve it by subtracting 3x from both sides, we end up with:

• 5 = -10.

This result clearly doesn't hold true; it’s a contradiction. Therefore, such an equation has no values for "x" that can satisfy it. These equations are termed as having no solution because they do not share any values that would make the equation hold true mathematically. Understanding no solution situations helps students identify when equations are inherently contradictory and impossible to satisfy.

Identity Equation

An "identity equation" is one that holds true for any value of the variable. These equations tend to simplify to a tautology, meaning they produce a statement that is always true, such as 0 = 0, or 5 = 5. Take, for instance, the equation: 2y + 3 = 2(y + 1) + 1. By expanding and simplifying both sides, we find:

• Left side: 2y + 3,
• Right side: 2y + 2 + 1,
• Simplifies to 2y + 3.

The simplified form for both sides is the same. Since both sides are equal for any value of "y", this equation is considered an identity. Understanding identity equations is crucial because, unlike equations with one or no solutions, identity equations represent general truths in mathematics, validating the balance in equations across all possible values of the variable.