Question

Solve the equation if possible. $$ -7 q-13=4-7 q $$

Short answer

The given equation has no solution.

Step-by-step solution

Combine like terms

The equation given is \( -7q - 13 = 4 - 7q \) . Here, there's a \(-7q\) term on both sides of the equation. So subtract \(-7q\) on both sides.

Simplify the Equation

On simplifying, it simplifies to \(-13 = 4\). This introduces an inconsistency, because -13 is not equal to 4.

Final Answer

Since -13 does not equals to 4, it results in an inconsistent statement. Given equation has no solution.

Key concepts

Combining Like Terms

Understanding how to combine like terms is a foundational skill in algebra. Like terms are terms in an equation that have the same variables raised to the same powers, even if their coefficients are different. In the equation
\[ -7q - 13 = 4 - 7q \]
we identify -7q on both sides as like terms. To simplify the equation, we aim to consolidate these terms.

Here's how you do it: subtract or add the like terms from both sides to effectively remove them from the equation. Mathematically, if you have a on both sides of the equation, you can perform a - a = 0, effectively canceling out the a term from each side. This step simplifies the equation and sets the stage for either solving the variable or reaching a conclusion about the solution's existence.

Simplifying Equations

Simplifying equations is the process of performing operations to rewrite the equation in its most basic form. This involves combining like terms, as discussed, and also performing addition, subtraction, multiplication, or division as required. Simplifying may involve moving terms from one side of the equation to the other or factoring.

• Always perform the same operation on both sides to maintain the equation's balance.
• The objective is to isolate the variable on one side to find its value.
• If variables cancel out and you are left with a true statement, such as \[ 0=0 \], the equation has infinitely many solutions.
• If the simplification leads to an absurdity, such as \[ -13 = 4 \], this signals that the equation may not have a solution.

For the given exercise, after combining like terms, we're left with a simplified statement that doesn't involve variables, triggering the next stage of evaluation.

No Solution

An equation that simplifies to a contradicting statement, such as \[ -13 = 4 \], signals that there is no solution. This indicates that no value of the variable will satisfy the equation — an important concept in algebra. This specific type of equation is known as an inconsistency. It's crucial to recognize these scenarios to avoid unnecessary attempts at solving what is unsolvable.

To analyze the possibility of a solution:

• First, simplify the equation as much as possible.
• After simplification, assess whether the remaining statement makes logical sense.
• If it is an impossible statement (an inequality involving only numbers), then declare that the equation has no solution.
• This is often represented by the symbol \[ \emptyset \] which stands for the empty set, signifying no value satisfies the equation.

In the context of the given exercise, our simplified equation -13 = 4 starkly presents no solution, as these numerical values will never be equal.