Question

Verify that the given value is a solution of the given equation. $$ 8 x-3=-11, x=-1 $$

Short answer

Question: Verify if x = -1 is a solution to the equation 8x - 3 = -11. Answer: Yes, x = -1 is a solution to the equation 8x - 3 = -11.

Step-by-step solution

Write down the given equation and given value of x

The given equation is: $$ 8x - 3 = -11 $$ And the given value of x is: $$ x = -1 $$

Substitute the given value of x into the equation

Replace x with -1 in the equation: $$ 8(-1) - 3 = -11 $$

Simplify the equation and check for equality

Calculate the left side of the equation: $$ -8 - 3 = -11 $$ The left side of the equation is equal to the right side, which shows that the given value of x is indeed a solution to the given equation.

Key concepts

Equation Verification

Equation verification is the process of determining if a given value satisfies the requirements of a particular equation. This means checking whether plugging the value into the equation makes it true. In the original exercise, we have the equation \(8x - 3 = -11\), and we need to verify if \(x = -1\) is a solution.
To do this, you substitute the value of \(x = -1\) into the equation and simplify the expression. If the left-hand side equals the right-hand side, then the given value is correct. This step ensures accuracy in solving algebraic equations and confirms whether our calculations are on the right track.
Here's a simple breakdown when verifying:

• Substitute the suggested solution into the equation
• Simplify both sides of the equation
• Check for equality: If both sides are equal, the verification is successful

Substitution in Algebra

Substitution in algebra involves replacing a variable with a given value. It’s a fundamental technique in solving equations. In the context of the exercise, substitution allows us to determine whether a specific value is a solution to an equation. When you replace \(x\) with \(-1\) in the equation \(8x - 3 = -11\), it looks like this:
\(8(-1) - 3 = -11\).

• The substitution aligns the algebraic expression with the numeric value provided
• It transforms a potentially complex expression into a simpler form
• Substitution provides a way to test solutions or, in other scenarios, to simplify expressions for further calculations

Remember, successful substitution keeps the integrity of the equation, and that's the key!

Linear Equation Simplification

Linear equation simplification is essential for breaking down complex expressions into their simplest form. Simplification helps us clearly see the relationship between quantities involved. In the exercise example, after substituting \(x = -1\), the expression becomes \(8(-1) - 3\).
This simplifies to \(-8 - 3 = -11\).
Simplification ensures clarity and correctness by highlighting equality or revealing errors.Here’s how simplification works:

• Perform mathematical operations step by step to reduce expressions
• Combine like terms and perform arithmetic operations
• Ensure equivalent forms to the original expressions, focusing on obtaining a true statement

Simplifying linear equations makes them much more approachable and facilitates better understanding of the relationship between variables and constants.