Problem 41 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ 3 x-7=2 x+8+4 x $$

Step-by-Step Solution

Answer

The equation has one solution: \(x = -5\).

Step 1: Simplify the equation

Combine like terms on both sides of the equation and express it in a simplified form. On the right side, combine \(2x\) and \(4x\) to get \(6x\), resulting in the simplified equation \(3x – 7 = 6x + 8\).

Step 2: Move terms containing x to one side and constant terms to the other side

Subtract \(3x\) from both sides and subtract \(8\) from both sides: \(–7 – 8 = 6x – 3x\), which simplifies to \(–15 = 3x\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 3. Doing so gives the solution \(x = -15 / 3 = -5\).

Key Concepts

One Solution

When an equation has exactly one solution, it means there is a single value for which the equation holds true. In our exercise, after simplifying and performing operations to isolate the variable, we found that the value of \( x \) that satisfies the equation is \(-5\).
Here are steps you often follow to derive a single solution:

• Combine similar terms so the equation is simplified.
• Isolate the variable by moving all instances of that variable to one side and constants to the other.
• Solve the resulting simple equation.

Obtaining one unique solution implies both sides of the equation can be made equal by adjusting for the one found \( x \)-value. It means the equation only intersects the line representing solutions at one distinct point.

No Solution

Equations with no solution signify that no real number can satisfy the equation. This usually happens when you have contradictions, such as ending with a false statement like \( 0 = 5 \).
For an equation to have no solution, follow these observations:

• Simplify the equation fully.
• If all variables cancel out and you're left with a false inequality or contradiction, it means no solution exists.

In simpler terms, a no-solution equation represents parallel lines that never intersect, hence can never be equal.

Identity

An identity in equations means every possible number is a solution. It is when both sides of the equation are the same after simplification. For example, an equation like \( x + 2 = x + 2 \) is true for all values of \( x \).
To identify an identity, keep these pointers in mind:

• Start by simplifying both sides thoroughly.
• If you end up with a tautology (a true statement like \( 0 = 0 \)), then the equation is an identity.

Identities are simply always true statements, reflecting complete overlaps of lines represented by the equation on any graph.