Problem 96 - Elementary Algebra Within Reach

Question

Determine whether (a) \(x=-1\) or (b) \(x=2\) is a solution of the equation. $$x+1.5=3.5$$

Step-by-Step Solution

Answer

\(x=-1\) is not a solution to the equation and \(x=2\) is a solution to the equation.

Step 1: Substitute (a) \(x=-1\) into the Equation

Substitute \(x=-1\) into the equation \(x+1.5=3.5\). This gives \(-1+1.5=3.5\), which simplifies to \(0.5=3.5\). This is not true, therefore \(x=-1\) is not a solution of the equation.

Step 2: Substitute (b) \(x=2\) into the Equation

Substitute \(x=2\) into the equation \(x+1.5=3.5\). This gives \(2+1.5=3.5\), which simplifies to \(3.5=3.5\). This is true, therefore \(x=2\) is a solution of the equation.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in solving algebraic equations, especially linear equations. In the context of our problem, substitution involves replacing the variable, in this case, \( x \), with a given value to see if it satisfies the equation. This method helps in identifying if a particular value is the correct solution to the equation.

When using substitution:

• Start with the equation provided and the potential values to test.
• Replace the variable with each given value one at a time.
• Simplify the equation following each substitution to check for equivalence on both sides.

This technique is straightforward and handy for not just validation but also finding solutions to more complex equations when paired with graphical methods or elimination. Substitution is a key player in algebraic problem-solving strategies.

Linear Equations

Linear equations are among the simplest types of equations you'll encounter in algebra. A linear equation is an equality that involves variables to the first power, such as \( x \), without any exponentiation beyond one. The general form of a linear equation involving one variable is \( ax + b = c \).

Key characteristics of linear equations include:

• They graph as straight lines in a coordinate plane.
• They have constant slopes, and solutions are unique unless represented by parallel lines.
• The operations needed to solve them are basic, covering addition, subtraction, multiplication, and division.

In our exercise, \( x + 1.5 = 3.5 \) is a simple linear equation. To solve such equations, one must isolate the variable, often by undoing the operations attached to it. Linear equations are foundational in understanding more advanced algebraic and calculus concepts.

Solution Verification

Solution verification is an essential step in solving equations. It ensures that the solution you have found is indeed correct. Verification involves substituting the solution back into the original equation and checking whether both sides of the equation are equal.

Here's how to verify a solution:

• Take the proposed solution and substitute it into the original equation.
• Simplify both sides of the equation using arithmetic operations.
• Check if the equality holds; if both sides of the equation are equal, then the solution is verified.

In our scenario, by substituting \( x = -1 \) and \( x = 2 \) into the equation \( x + 1.5 = 3.5 \), only by using \( x = 2 \) did we get a valid equation: \( 3.5 = 3.5 \). Verification reassures that tasks are completed correctly, thereby avoiding any misunderstanding in mathematical solutions.