Problem 25 - Algebra 1

Question

Is the number given a solution of the equation? $$2 y+8=4 y-2 ; 5$$

Step-by-Step Solution

Answer

Yes, the number 5 is a solution to the equation.

Step 1: Substitution

Substitute the given number 5 in place of \(y\) in the equation. Therefore, the equation now looks like this: \(2*5+8=4*5-2\).

Step 2: Simplify Both Sides

Simplify both sides of the equation separately. On the left side, \(2*5+8\) gives 18. On the right side, \(4*5-2\) gives 18.

Step 3: Compare the results

Upon comparison, it can be seen that both the left side and the right side of the equation simplify to the same number, 18. Therefore the number 5 is indeed a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a critical skill in algebra that involves replacing variables with given numbers to determine if those numbers satisfy an equation. This method is commonly used to verify solutions or to solve systems of equations.

Let's apply this to an exercise: Is the number given a solution of the equation \(2y + 8 = 4y - 2; 5\)? To utilize the substitution method, you would replace \('y'\) with the number 5. The equation would then be \(2*5 + 8 = 4*5 - 2\). This step transforms the equation into a numerical expression, eliminating the variable and allowing for direct evaluation.

Simplifying Expressions

Simplifying expressions is an essential process that involves reducing complex algebraic expressions into simpler, more manageable forms. This makes it easier to handle the arithmetic and leads to more straightforward evaluation or further manipulation of equations.

In our example with the substituted value of 5, simplifying the expressions on both sides of the equation is the next step. This means performing the operations indicated: \((2*5 + 8)\) simplifies to 18 and so does \((4*5 - 2)\). The simplification process clearly shows that both sides of the equation are equal when simplified independently, suggesting that the initial substitution was correct.

Evaluating Solutions

Evaluating solutions involves analyzing the results obtained from simplifying expressions with the substituted values to conclude whether those values make the original equation true.

This step often results in the verification of a possible solution. In our scenario, after substitution and simplification, the expressions on both sides of the equation equaled 18, indicating that the evaluation process confirms the number 5 as a valid solution to the equation \(2y + 8 = 4y - 2\). When both sides match after the simplification, the evaluated number satisfies the original equation, which is the main goal of the evaluation phase.