Problem 53 - Algebra 1

Question

Check whether the number is a solution of the equation or the inequality. \(7 y+2=4 y+8 ; 2\)

Step-by-Step Solution

Answer

Yes, the number 2 is a solution of the equation \(7 y+2=4 y+8\).

Step 1: Substitution Step

Substitute the given number for 'y' in the equation \(7 y+2=4 y+8\). Thus, the equation becomes \(7*2+2 = 4*2+8\). Simplifying both sides yields 16 = 16.

Step 2: Comparison Step

Compare both sides of the equation. If they are equal, then the given number is a solution of the equation. In this case, both sides are equal (16 = 16), confirming that 2 is indeed a solution of the equation \(7 y+2=4 y+8\).

Key Concepts

Solution Verification

When we talk about solution verification, we mean checking if a particular value satisfies a given equation or inequality. This is important because it ensures the accuracy of our solution.
To verify a solution, we substitute the given number into every occurrence of the variable within the equation or inequality. For instance, in the equation \(7y + 2 = 4y + 8\), we substitute 2 for \(y\). This substitution transforms the equation into \(7 \times 2 + 2 = 4 \times 2 + 8\). After simplifying both sides, we find that both equations equate to 16.
The final step is comparison. If both sides of the equation are equal after substitution, the number is indeed a solution to the equation. Thus, solution verification confirms whether a given number truly satisfies the equation or not.

Substitution Method

The substitution method is a fundamental technique used to solve equations, especially when verifying solutions or dealing with systems of equations. It involves replacing a variable with a given number to simplify the equation for easier solving.
In our example, we start by substituting 2 for \(y\) in the equation \(7y + 2 = 4y + 8\). This step changes the equation into numeric terms: \(7 \times 2 + 2\) for one side and \(4 \times 2 + 8\) for the other side. By converting the equation into numbers, we can easily compare the results.
This method is not only used for verifying solutions, but it’s also useful in solving systems of equations. By substituting known values, it becomes simpler to find unknowns and verify if solutions work in multiple equations. The substitution method is a robust tool for simplifying complex problem-solving scenarios.

Equations and Inequalities

Equations and inequalities form the basis of many mathematical problems and solutions. An equation is a statement that expresses the equality between two expressions, such as \(7y + 2 = 4y + 8\).
On the other hand, inequalities show the relationship where one expression is greater than or less than another. Solving equations and inequalities often involves finding the value of the unknown variable that makes the statement true.

• For equations, the goal is to find a value that makes both sides equal.
• Inequalities require finding a range of values that satisfy the relationship, such as \(x > 5\) or \(3x + 2 \leq 14\).

Understanding how to handle both equations and inequalities is crucial since they each have unique rules and methods for solving.
This foundational knowledge enables problem solvers to tackle a variety of mathematical challenges effectively.