Question

Verify that the given value is a solution of the given equation. $$ 3 x-7=-28, x=-7 $$

Short answer

Question: Verify if x = -7 is a solution to the equation 3x - 7 = -28.

Step-by-step solution

Write down the original equation and the given value of x

The original equation is \(3x - 7 = -28\), and the given value of x is \(x = -7\).

Substitute the given value of x into the equation

Replace x with -7 in the equation, like this: \(3(-7) - 7 = -28\)

Simplify the left-hand side of the equation

Multiply the 3 by the -7 and subtract 7 from the result: \((-21) - 7 = -28\)

Compare the simplified left-hand side with the right-hand side

Check if the simplified left-hand side equals the right-hand side: \(-28 = -28\)

Confirm the solution

Since the left-hand side equals the right-hand side after substituting the given value of x, we can confirm that \(x = -7\) is indeed a solution of the given equation \(3x -7 = -28\).

Key concepts

Linear Equations

Linear equations are a fundamental concept in algebra. They represent equations of the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable to be solved. These equations form a straight line when graphed on a coordinate plane.
For example, in the exercise given, the equation is \(3x - 7 = -28\). This is a linear equation because it has the variable \(x\) raised to the first power and only involves addition, subtraction, and multiplication. Linear equations can be easily solved through systematic algebraic techniques, which makes them one of the earliest introduced topics in algebra classes. Here are a few key characteristics of linear equations:

• They have one or two variables.
• Their graph is always a straight line.
• There's a constant rate of change between the variables.

Solving linear equations is a critical skill because it lays the groundwork for more advanced algebra topics.

Algebraic Manipulation

Algebraic manipulation involves a series of steps to simplify equations or expressions to either solve for a variable or verify a solution. This manipulation is vital because it helps to isolate the variable in question and test if a specific value satisfies the equation.
In our example, the equation \(3x - 7 = -28\) necessitates several manipulative steps. We start by substituting the potential solution value, which is \(x = -7\). When substituted into the equation, it becomes \(3(-7) - 7 = -28\). The aim here is to simplify the left-hand side to verify that it equals the right-hand side. Here are the practical steps performed in algebraic manipulation:

• Substituting known values into the equation.
• Simplifying expressions by performing arithmetic operations (addition, subtraction, multiplication, etc.).
• Maintaining the equality by performing the same operations on both sides of the equation when needed.

This process is essential for both solving and verifying if a particular solution works for an equation.

Solution Verification

Solution verification is the process of checking whether a specific value satisfies an equation. It's essential in problem-solving to ensure that the solution derived is indeed correct.
In the given exercise, after substituting \(x = -7\) into the equation and simplifying, we ended with the statement \(-28 = -28\). Solution verification assures us that the mathematical manipulations were conducted correctly and the value substituted finds consistency in the equation.
Here's a concise summary of how solution verification works in linear equations:

• Substitute the given value into the original equation.
• Simplify the side of the equation with the variable to see if it matches the opposite side.
• If both sides of the equation are equal after substitution and simplification, the solution is verified as correct.

Verification is a critical step in solving equations because it provides confidence that the solution satisfies the original problem.