Problem 28 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF EQUATIONS. Check to see if the given value of the variable is or is not a solution of the equation. $$ 4 c+2=10 ; c=2 $$

Step-by-Step Solution

Answer

Yes, \(c = 2\) is a solution to the equation.

Step 1: Identify the Equation

Here, the given equation is \(4c + 2 = 10\). We are asked to check if \(c = 2\) is the solution of the equation.

Step 2: Substitute the Given Value

Substitute the given solution \(c = 2\) into the equation. Replace every occurrence of 'c' in the equation with '2'. So, the equation becomes \(4*2 + 2\).

Step 3: Check the Equality

Solve the equation on the left hand side, \(4*2 + 2 = 10\). If the resultant value equals the right side of the equation, then \(c = 2\) is indeed the solution. In this case, \(4*2 + 2 = 8 + 2 = 10\), which is equal to the right hand side of the equation.

Key Concepts

Substitute the Given Value

When tackling algebraic equations, one of the most fundamental skills is the ability to substitute a given value into an equation. This process is used to determine if that value is a solution. In practical terms, substitution involves replacing the variable in the equation with the given number and simplifying.

For example, if you have the equation \(4c + 2 = 10\) and you want to check if \(c = 2\) is a solution, you would follow these steps:

• Identify the variable in the equation, which is 'c'.
• Replace 'c' with the number 2 in the equation, resulting in \(4 \times 2 + 2\).
• Simplify this expression by performing the multiplication and addition to see if you get the other side of the equation, which is 10 in this case.

Verify Equality in Algebra

Verification of equality is the process used to confirm that both sides of an equation are equivalent once a value has been substituted for the variable. This is a critical step in validating the solution to an equation.

Continuing with our example, after substituting \(c = 2\) into the original equation, we calculate the left-hand side, obtaining \(8 + 2\), which simplifies to 10. Now, it's time to compare this with the right-hand side. Since both sides equal 10, you have verified that the equation is balanced, and thus \(c = 2\) is indeed a correct solution.

To emphasize:

Solve Algebraic Equations

Solving an algebraic equation means finding all values that make the equation true. It involves isolating the variable and determining what numbers satisfy the equation.

To solve the equation \(4c + 2 = 10\), you would perform the following steps:

• Subtract 2 from both sides of the equation to get \(4c = 8\).
• Divide both sides by 4, concluding that \(c = 2\).

By solving the equation systematically, we've found the solution that verifies the original equality. Importantly, this solution is not simply assumed; it is the result of a logical, step-by-step procedure that confirms its validity. Additionally, considering multiple solutions or a 'trial and error' approach can also be part of solving algebraic equations, especially when they become more complex.

The ability to solve equations is essential for success in algebra and forms the basis for more advanced topics in mathematics.