Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation using both the addition and multiplication properties of equality. Check proposed solutions. $$12=4 z+3$$

Step-by-Step Solution

Answer

The solution to the equation is \(z = 9/4\) or \(z = 2.25\) in decimal form.

Step 1: Isolate the z-term

Subtract 3 from both sides of the equation, which gives us:\(12 - 3 = 4z + 3 - 3\), simplifying to: \(9 = 4z\).

Step 2: Solve for z

To isolate z, divide both sides of the equation by 4: \(9/4 = 4z/4\). This simplifies to \(z = 9/4\) or \(z = 2.25\) in decimal form.

Step 3: Check the solution

Substitute \(z = 9/4\) into the original equation: \(12 = 4*(9/4) + 3 \). Simplifying the right-hand side gives exactly 12, confirming that the solution is correct.

Key Concepts

Addition Property of Equality

When solving linear equations, the addition property of equality is a key tool. It states: If you add or subtract the same number from both sides of an equation, the equality remains.
This property helps us manipulate equations without upsetting their balance. In our exercise, you see this applied when we subtract 3 from both sides. Starting with the equation:

• \(12 = 4z + 3\)

We subtract 3 from both sides to keep the equation balanced:

• \(12 - 3 = 4z + 3 - 3\)

This simplifies the equation to:

• \(9 = 4z\)

Similar steps help rearrange terms and simplify expressions in various equations. Remember, maintaining balance is key to solving effectively. Keep subtracting or adding consistently to both sides.

Multiplication Property of Equality

After applying addition or subtraction, we often use the multiplication property of equality to further solve the equation. It states: Multiplying or dividing both sides of an equation by the same non-zero number keeps the equation balanced.
Using our simplified equation:

• \(9 = 4z\)

We need to isolate \(z\). To do this, we divide by 4, which is the coefficient of \(z\):

• \(\frac{9}{4} = \frac{4z}{4}\)

This division isolates \(z\):

• \(z = \frac{9}{4}\)

Adopting the multiplication property this way helps us find \(z\)'s value. Notice here, dividing both sides allowed the equation to stay true, leading straightforwardly to the correct answer. Practice converting multiplication or division to undo operations and isolate variables.

Solution Verification

Once we find a potential solution, checking it ensures accuracy. Solution verification means substituting the value back into the original equation and verifying both sides equal.
For instance, with \(z = \frac{9}{4}\), substitute back into:

• \(12 = 4\left(\frac{9}{4}\right) + 3\)

Simplify the right side:

• \(12 = 9 + 3\)
• \(12 = 12\)

Both sides match, confirming \(z = \frac{9}{4}\) is correct. Verification is crucial; it eradicates potential calculation mishaps and reinforces understanding. Performing checks whenever finding solutions is a good habit to cultivate.

Isolation of Variables

The crux of solving linear equations is isolating the variable. This means manipulating the equation until the variable stands alone on one side.
Start with given equation:

• \(12 = 4z + 3\)

Subtract 3 from both sides to remove constant terms with:

• \(9 = 4z\)

Then, divide by 4 to detach the variable:

• \(z = \frac{9}{4}\)

When a variable is isolated, it translates to finding its value. Use these steps consistently. It simplifies the problem, turning a complex equation into an easily discernible solution. Practice this with different variables and constants to bolster your equation-solving skills.