Question

Verify that the given value is a solution of the given equation. $$ 11 x-1=10, x=1 $$

Short answer

Question: Verify if x = 1 is a solution to the equation 11x - 1 = 10. Answer: Yes, x = 1 is a solution to the equation 11x - 1 = 10.

Step-by-step solution

Write down the given equation and value of x

The given equation is: $$ 11x-1=10 $$ And the given value of x is: $$ x = 1 $$

Substitute the value of x into the equation

Now, substitute the given value of x (x = 1) into the equation: $$ 11(1) - 1 = 10 $$

Simplify the equation

Simplify the equation by performing the multiplication and subtraction: $$ 11 - 1 = 10 $$

Check if both sides of the equation are equal

Now, check if both sides of the equation are equal: $$ 10 = 10 $$ Since both sides of the equation are equal, this verifies that the given value of x (x = 1) is a solution to the given equation (11x - 1 = 10).

Key concepts

Substitution Method

The substitution method is a useful technique for solving equations, especially when you have a known value for one of the variables. It involves replacing the variable in the equation with its given or known value to simplify and solve the equation.
Let's walk through how to use this method. Given the equation \(11x - 1 = 10\), and the value \(x = 1\), we substitute "1" into "x." This changes the equation to \(11(1) - 1 = 10\).

• Identify the given equation and value for the variable.
• Replace the variable with the given value.
• Perform the arithmetic operations with the substitute value.

By substituting, you transform the equation into a simple arithmetic problem that can easily be verified.

Equation Verification

Once you have substituted the variable's value into the equation, the next step is to verify if it makes the equation true. This involves simplifying the equation and checking if both sides are equal.
Continuing with our earlier example, after substituting \(x = 1\), we simplify the equation: \(11(1) - 1 = 10\) results in \(10 = 10\). If this is true, the substitution was correctly done. If both sides match, it confirms that the value is indeed a solution for the equation.

• Perform necessary arithmetic to simplify.
• Compare both sides of the equation after simplification.
• If equal, the value is verified as a correct solution.

Equation verification ensures the accuracy and validity of the solution.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make it easier to solve or interpret. This skill is vital when working with equations because it allows you to isolate variables, simplify complicated expressions, and verify solutions.
In our example, the equation \(11x - 1 = 10\) was manipulated by performing arithmetic operations once the substitution was made. This involved simplifying \(11(1) - 1\) to just \(10\), illustrating how basic manipulation can solve or verify equations.

• Perform operations like addition, subtraction, multiplication, and division.
• Focus on simplifying each part of the equation.
• Keep both sides of the equation balanced.

Understanding algebraic manipulation is crucial not only in confirming solutions but also in rearranging equations to reveal their meaning or connections.