Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To __ a proposed solution means to substitute it into the original equation and see whether a true statement results.

Step-by-Step Solution

Answer

To verify a proposed solution means to substitute it into the original equation.

Step 1: Understanding the Goal

When you have a proposed solution to an equation, you need to check whether that solution actually satisfies the equation. This involves substituting the proposed values into the equation and verifying if the equation holds true.

Step 2: Substitution Process

To verify the proposed solution, substitute the values given as the solution into the equation. This means replacing the variable(s) in the equation with the proposed solution value(s).

Step 3: Check for Validity

After substituting the values, perform the necessary arithmetic operations to simplify the expression. Check the final result to see if both sides of the equation are equal, confirming whether a true statement is achieved.

Step 4: Conclusion

If the simplified equation results in a true statement, the proposed solution is valid for the equation. If not, the solution proposed does not satisfy the equation.

Key Concepts

Substitution Method

In algebra, the substitution method is a technique used to confirm whether a proposed solution to an equation is correct. Suppose you have an equation and a set of proposed solution values for the variables in it. You apply the substitution method by taking these values and inserting them into the equation wherever the variables occur. This simple yet effective process is like solving a puzzle by trying different pieces.

• Identify the variables in the equation and the proposed solution values.
• Replace (or "substitute") each variable with its corresponding proposed value.
• Perform any computations needed to fill in the blanks left by the substitution.

After substituting, you continue to check if the outcome suits both sides of the equation, which brings us to the next concept: validity of equations.

Validity of Equations

When you substitute proposed values into an equation, you must verify whether the completed math expression remains balanced. This is what we call checking the validity of an equation. Think of this as making sure that a seesaw stays level by keeping equal weight on either side.

• After substitution, perform any necessary calculations on both sides of the equation.
• Compare the mathematical expressions on both sides. Check if the left side equals the right side.
• If equal, the proposed solution is valid. If not, the solution is not valid for the equation.

Checking validity is crucial because an inaccurate solution won’t solve the problem correctly. This method ensures that the proposed solution actually represents the values needed to satisfy the equation fully.

Arithmetic Simplification

Arithmetic simplification is the last, but one of the most important steps in verifying solutions. Once you have substituted the values and set up your equation, simplifying will allow you to clearly see if both sides match. Simplification involves carrying out all the arithmetic operations like addition, subtraction, multiplication, or division with the substituted values.

• Perform operations as per the order of operations (PEMDAS/BODMAS).
• Simplify each side of the equation step-by-step: combine like terms and simplify complex fractions or expressions.
• Once fully simplified, contrast the two sides of the equation to verify equality.

Simplifying correctly is essential because it transforms a potentially complicated expression into something much easier to compare and confirm. This is the final checkpoint in determining the validity of your proposed solution.