Problem 36 - Elementary Algebra

Question

Is the given value a solution to the linear equation? $$ 2 x+4=3 x-2 ; x=-1 $$

Step-by-Step Solution

Answer

No, \( x = -1 \) is not a solution; it does not satisfy the equation.

Step 1: Substitute the value

First, substitute the given value of \( x = -1 \) into both sides of the equation to determine if it satisfies the equation. This means replacing \( x \) with \( -1 \) in both sides of the equation \( 2x + 4 = 3x - 2 \).

Step 2: Solve the left side

Substitute \( x = -1 \) in the left side of the equation: \[2(-1) + 4 = -2 + 4 = 2\]

Step 3: Solve the right side

Substitute \( x = -1 \) in the right side of the equation:\[3(-1) - 2 = -3 - 2 = -5\]

Step 4: Check the equality

Compare the results from both sides of the equation. The left side equals \( 2 \), while the right side equals \( -5 \). Since \( 2 eq -5 \), \( x = -1 \) is not a solution to the equation.

Key Concepts

Solution Verification

When faced with a linear equation, it's crucial to check if a proposed solution holds true. This process, called solution verification, ensures that the value we substitute for the variable satisfies both sides of the equation. Solution verification helps confirm the integrity of our mathematical work.

In practice, verification involves substituting the proposed solution into the original equation. For example, with the equation \( 2x + 4 = 3x - 2 \) and a proposed solution of \( x = -1 \), we substitute \( -1 \) into the equation.

We then independently calculate the results for each side of the equation:

• The left side yields \( 2(-1) + 4 = 2 \).
• The right side yields \( 3(-1) - 2 = -5 \).

Since these values do not match, \( x = -1 \) is not a valid solution. Solution verification is a key step to ensure accuracy in solving linear equations.

Substitution Method

The substitution method is a fundamental technique used to check potential solutions for an equation. When we apply this method, we replace the variable of an equation with a given number, essentially testing to see what results from this replacement.

Consider our linear equation \( 2x + 4 = 3x - 2 \). By using the substitution method with \( x = -1 \):

• Substitute \( x = -1 \) into the equation.
• The left side becomes \( 2(-1) + 4 \).
• The right side becomes \( 3(-1) - 2 \).

After performing the arithmetic, we compare both sides to determine if they are equal. This method helps verify or reject proposed solutions and is especially powerful for equations involving one variable.

Algebraic Expressions

Algebraic expressions form the basis of equations in algebra, providing a structure for solving problems. An algebraic expression consists of variables, coefficients, and constants combined using operations like addition and multiplication.

In our equation \( 2x + 4 \) and \( 3x - 2 \), the terms are examples of algebraic expressions. Each term combines a coefficient (like 2 or 3) and the variable \( x \). The numbers 4 and -2 are constants added to or subtracted from these variable terms.

Understanding algebraic expressions is crucial for simplifying and evaluating equations. By manipulating these expressions, we can isolate variables, solve for unknowns, and verify whether potential solutions are valid. Recognizing and working with these components helps in depths of algebraic problem-solving.