Problem 116 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 3-z=-2 $$

Step-by-Step Solution

Answer

The solution to the equation is \(z = 5\).

Step 1: Identify the Equation

Our given equation is \(3 - z = -2\). We need to find the value of \(z\).

Step 2: Isolate the Variable

To solve for \(z\), we need to isolate it on one side of the equation. Our equation is \(3 - z = -2\).To isolate \(z\), first subtract 3 from both sides of the equation:\[3 - z - 3 = -2 - 3\]

Step 3: Simplify Both Sides

Simplify both sides after subtraction:Left side: \(3 - 3 = 0\), so we have \(-z\) on the left side.Right side: \(-2 - 3 = -5\).Thus, the equation becomes \(-z = -5\).

Step 4: Solve for the Variable

To solve for \(z\), multiply both sides of the equation by \(-1\) to eliminate the negative sign:\[-z \times (-1) = -5 \times (-1)\]This simplifies to:\[z = 5\]

Step 5: Verify the Solution

Substitute \(z = 5\) back into the original equation to check our solution:Check: \(3 - z = 3 - 5 = -2\).The left side equals \(-2\), which matches the right side of the original equation. Thus, \(z = 5\) is correct.

Key Concepts

Isolating the Variable

In solving linear equations, isolating the variable is a crucial first step. It means maneuvering the equation to have the variable on one side by itself. In the example equation, \(3 - z = -2\), our goal is to solve for \(z\). Start by moving all other terms to the opposite side of the equation. This is achieved by subtracting 3 from both sides, leading to:

• \(3 - z - 3 = -2 - 3\)
• Which simplifies to \(-z = -5\).

This operation of inverse actions, such as subtracting or adding, helps in effectively meeting the goal of "looking at the variable alone" without other numbers or terms attached to it. Remember, whatever operation you perform on one side, you must equally apply it to the other side to keep the equation balanced.
To eliminate the negative sign in front of the variable, multiply the entire equation by \(-1\) to finally express \(z\) alone:

• \(-z \times (-1) = -5 \times (-1)\)
• This simplifies to \(z = 5\).

By isolating \(z\) and simplifying properly, you've effectively found the solution.

Verification of Solutions

Verification is the process of ensuring that the solution you found is correct. It's a critical step that confirms the integrity of the answer in solving linear equations. For our example, substituting the value \(z = 5\) back into the original equation should give you an equal left and right side. Let's substitute and check:

• Original equation: \(3 - z = -2\)
• Substitute \(z = 5\): \(3 - 5 = -2\)
• 0 equals \(-2\), which shows that both sides are equal.

Upon substitution, if both sides match, you verify that your solution, \(z = 5\), is accurate. This process not only affirms correct calculations but also helps develop confidence in manipulating and solving equations accurately.

Simplification of Expressions

Simplification is reducing the complex parts of an equation to make it easier to understand or solve. This process takes place right after performing operations such as subtraction, addition, multiplication, or division while isolating the variable. In our case, after subtracting 3 from both sides, the left side became \(3 - 3 = 0\), simplifying to \(-z\), while the right side resulted in:

• \(-2 - 3\), which simplifies to \(-5\).
• Thus, we had \(-z = -5\).

This clear reduction into simpler parts helps navigate directly toward the answer. Additionally, correcting signs, breaking down, or rearranging terms align equations nicely, preparing them for the next solving steps.
Simplification ensures you don't get stuck on either messing up the algebraic cleanliness or convoluted calculations, essential for accuracy and efficiency in solving equations.