Problem 52 - Elementary Algebra

Question

Solve. $$ -4-7 a=24 $$

Step-by-Step Solution

Answer

The solution is \(a = -4\).

Step 1: Isolate the Variable Term

To solve the equation \[-4 - 7a = 24\]we need to isolate the term with the variable, \(a\). Start by adding \(4\) to both sides of the equation to eliminate the \(-4\) from the left side:\[-4 - 7a + 4 = 24 + 4\]Simplifying both sides gives:\[-7a = 28\]

Step 2: Solve for the Variable

Now that we have \[-7a = 28\]we can solve for \(a\) by dividing both sides of the equation by \(-7\):\[a = \frac{28}{-7}\]This simplifies to:\[a = -4\]

Key Concepts

Variable Isolation

Isolating the variable is a crucial step when dealing with linear equations. In our equation, \(-4 - 7a = 24\), we focus on isolating the term with \(a\) to one side of the equation. Set your sights on separating the variable term from constants.

• Look for operations attached to the variable, such as addition or subtraction, and start reversing them.
• For instance, in \(-4 - 7a = 24\), the \(-4\) is subtracted; hence, to neutralize it, we add \(4\) on both sides.

By performing this operation, the equation simplifies to \(-7a = 28\). Once you've isolated the variable term, solving for the variable becomes straightforward, as you simply need to eliminate the coefficient attached to the variable.

Solving Equations

Solving equations like \(-7a = 28\) involves finding the value of the variable that makes the equation true. Once you've isolated the variable term, the next step is to solve for \(a\). Here are the steps to solve:

• Look at the term with the variable (\(-7a\) in our case) and identify the coefficient of the variable. The coefficient here is \(-7\).
• You aim to have \(a\) by itself, which means getting rid of \(-7\). This is done by dividing both sides of the equation by \(-7\).
• Perform the division: \(a = \frac{28}{-7}\).

This results in \(a = -4\). By following these simple steps, you achieve a clean and clear solution to the equation. Remember, whatever operation you perform on one side of the equation, do it equally on the other to maintain balance.

Negative Coefficients

Understanding negative coefficients is key when solving equations. A negative coefficient is simply a coefficient with a negative sign, like the \(-7\) in our equation \(-7a = 28\).

• A negative coefficient indicates that the variable is "inversed" in a sense; it leads to an opposite direction when solving.
• To neutralize its effect, you often use division or multiplication, which transforms a negative into a positive effect.
• In our equation, dividing by \(-7\) reverses the negative, simplifying the calculation to finding positive \(a\).

These balancing acts ultimately guide you to the solution \(a = -4\), deftly steering you to the correct answer with confidence in handling the negativity.