Question

Verify that the given value is a solution of the given equation. $$ 7 x+7=7, x=0 $$

Short answer

Answer: Yes, x=0 is a valid solution to the equation 7x+7=7.

Step-by-step solution

Substitute the given value of x into the equation

We will replace x with its given value (0) in the equation 7x+7=7, like so: $$ 7(0)+7=7 $$

Simplify the equation

Consult your order of operations to compute the result. First, multiply 7 times 0 and then add 7, which simplifies to: $$ 0+7=7 $$

Determine if the equation holds true

Examine the simplified equation: 0+7=7. We can see that the left side (0+7) equals 7 and that the right side also equals 7. Therefore, we can conclude that: $$ 7=7 $$

Confirm that the given value of x is a valid solution

Since the equation holds true after substituting the given value of x (x=0), it confirms that x=0 is indeed a solution to the given equation 7x+7=7.

Key concepts

Solution Verification

When solving algebraic equations, it's crucial to verify that a solution is correct. This process ensures that the proposed solution satisfies the original equation. To verify a solution, follow these steps:

• First, substitute the value back into the original equation. In our exercise, where the equation is \(7x + 7 = 7\) and the proposed solution is \(x = 0\), substitute for \(x\).
• Next, simplify the equation using basic arithmetic to see if both sides are equal.
• Finally, compare the left-hand side of the equation with the right-hand side. If they are equivalent, as in this example where both sides simplify to 7, the solution is verified.

Verification is a critical step because it detects any errors in your calculations or assumptions before you accept a solution as valid. It's a simple yet effective way to ensure accuracy in mathematical problem-solving.

Substitution Method

The substitution method is a straightforward technique used to check solutions by replacing variables with numbers. This is especially useful in problems like ours, where you are given potential solutions to an equation. Here's a breakdown of how it works:

• Identify the given value for the variable in the equation. For instance, \(x = 0\) in the equation \(7x + 7 = 7\).
• Replace the variable with this value in the equation. This step transforms the equation from algebraic form to a numeric form, making it easier to solve.
• Perform any necessary arithmetic operations, as dictated by the equation, to simplify it and verify its truth.

The substitution method is essential not only because it helps in checking solutions, but also because it enhances understanding of how changes in the variable affect the equation. This method provides visual clarity on how each component of an equation contributes to the overall solution process.

Order of Operations

Order of operations is a fundamental principle in mathematics that dictates the correct sequence to apply arithmetic operations in more complex calculations. Following the order of operations eliminates ambiguity and ensures consistent results. Here’s a simple guide to the order of operations, often remembered by the acronym PEMDAS:

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Next, calculate any exponents.
• Multiplication and Division: Perform these operations from left to right.
• Addition and Subtraction: Finally, execute these steps, again moving from left to right.

In our example equation \(7x + 7 = 7\) with \(x = 0\), we first multiply \(7\) and \(0\) according to the multiplication step, resulting in \(0+7\). Subsequently, we execute the addition \(0+7\), which equals 7. Adhering to the order of operations ensures proper evaluation of the equation, confirming that the side values indeed equal, thus validating the solution.