Problem 2 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$4 x+3=7 ; 1$$

Step-by-Step Solution

Answer

Yes, 1 is a solution to the equation.

Step 1: Substitute the Solution

To check if 1 is a solution to the equation, substitute 1 for \(x\) in the equation \(4x + 3 = 7\). So, replace \(x\) with 1 and write it as:\(4(1) + 3\)

Step 2: Simplify the Expression

Next, simplify \(4(1) + 3\) by performing the multiplication:\(4 \times 1 = 4\)So the expression becomes \(4 + 3\).

Step 3: Perform Addition

Now add 3 to the result from the multiplication step:\(4 + 3 = 7\).This equals the right side of the equation \(7\).

Step 4: Conclusion

Since both sides of the equation are equal, 1 is indeed a solution to the equation \(4x + 3 = 7\).

Key Concepts

Solution Verification

Verifying a solution to an equation involves checking if a given value satisfies the mathematical condition stated in the equation. In other words, it's about confirming whether the number we have makes the equation true. This is often the final step after solving an equation, where you simply check that the left side and right side of the equation are equal when you substitute your solution back into the equation.

• First, identify the equation and the suggested solution value.
• Substitute the solution value into the equation.
• Simplify both sides if necessary.
• If both sides are equal, the solution is verified.

This helps in ensuring that no errors were made during the calculations and that the solution is correct and valid. Verifying is a simple task but crucial for understanding and confidence in your solution.

Substitution Method

The substitution method is a technique used to find the solution to an equation or system of equations. It involves replacing a variable with its equivalent value or expression. This method is highly effective for equations that can be rearranged or when verifying solutions by substitution.

• Start by identifying the variable you need to substitute.
• Replace this variable with the proposed solution or its equivalent expression.
• Proceed with simplifying or resolving the equation after the substitution.

This method often simplifies complex problems and allows for easy verification of proposed solutions, as seen in checking whether a given number is a solution to the equation. Through substitution, you actively engage with the problem, enhancing your understanding of how solutions interact with the original equation.

Simplification Process

The simplification process in mathematics refers to the reduction of expressions into their simplest or most understandable form. When applied to linear equations or substitutions, simplification makes it easier to compare the sides of an equation.

• Begin by performing any operations within parentheses.
• Carry out all multiplications and divisions, moving from left to right.
• Finally, handle additions and subtractions.

In verifying an equation, simplification helps ensure both sides of the equation match exactly when substitution has been applied. It reduces the possibility of errors and provides a clear path to the solution. Proper simplification is crucial for precise mathematical problem-solving and enhances comprehension of mathematical operations.