Problem 85 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ -4 x-5=-3 $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\).

Step 1: Isolate the Variable Term

Begin by isolating the term with the variable. To do this, add 5 to both sides of the equation to eliminate the constant term on the left side. This will result in:\[-4x - 5 + 5 = -3 + 5\]Which simplifies to:\[-4x = 2\]

Step 2: Solve for the Variable

To solve for \(x\), divide both sides of the equation by -4 to isolate \(x\):\[\frac{-4x}{-4} = \frac{2}{-4}\]The equation simplifies to:\[x = -\frac{1}{2}\]

Key Concepts

Isolating the Variable

Whenever you face a linear equation, the first task is usually to isolate the variable you want to solve for. This process involves rearranging the equation so that the term containing the variable is by itself on one side of the equation. In our example, the variable is \( x \). Our initial equation is \(-4x - 5 = -3\). We need to get \(-4x\) by itself. To do that, we look for any constants on the same side as our variable term. Here, \(-5\) is the constant we want to eliminate from the left side of the equation.

• Add or subtract the constant from both sides to maintain the equality.
• In this case, add 5 to both sides: \(-4x - 5 + 5 = -3 + 5\).
• The equation simplifies to \(-4x = 2\).

Now \( x \) is part of an equation much easier to resolve. The key is maintaining balance: what you do to one side, do to the other.

Solving for the Variable

Once you've isolated the term with the variable, the next step is to solve for the variable itself. We are left with the equation \(-4x = 2\). Here, \( x \) is still paired with \(-4\), so we need to undo this multiplication by dividing.

• Divide both sides of the equation by \(-4\) to separate \( x \).
• This looks like: \( \frac{-4x}{-4} = \frac{2}{-4} \).
• The equation simplifies to \( x = -\frac{1}{2} \).

By dividing by \(-4\), you isolate \( x \) entirely, giving you its value. Each step brings you closer to uncovering what \( x \) equals, which is the primary goal of solving equations.

Linear Equation Steps

Solving linear equations is often a step-by-step process. Each step builds upon the last, moving you closer to finding the value of the unknown variable. There's a logical flow to these steps that makes solving equations straightforward.

• Begin by identifying the variable and any constants on the same side.
• Use addition or subtraction to eliminate constants and isolate the variable term.
• Consider the operations paired with the variable, like multiplication or division, and use the opposite operation to solve for the variable.

In our exercise, we initially added 5 to both sides to eliminate the constant. Then, we divided by \(-4\) to complete the isolation of \( x \). Following these structured steps can make even complicated equations manageable and ensures that no aspect of the equation is overlooked.