Problem 34 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 5(3+z)-(8 z+9)=-4 z $$

Step-by-Step Solution

Answer

The solution is \(z = -6\).

Step 1: Distribute and Simplify

Start by distributing the 5 in the term \(5(3+z)\). This results in \(15 + 5z\). Next, the equation becomes \(15 + 5z - (8z + 9) = -4z\). Simplify the expression by distributing the negative sign across \( (8z + 9) \), which gives us \(15 + 5z - 8z - 9 = -4z\). After combining like terms, the equation simplifies to \(6 - 3z = -4z\).

Step 2: Isolate the Variable

To isolate \(z\), we need to get all \(z\) terms on one side. Add \(3z\) to both sides of the equation: \(6 - 3z + 3z = -4z + 3z\), simplifying to \(6 = -z\).

Step 3: Solve for the Variable

Now, solve for \(z\) by multiplying or dividing both sides by -1 to isolate \(z\). Doing so gives us \(z = -6\).

Step 4: Check the Solution

Substitute \(z = -6\) back into the original equation to verify the solution. The original equation is \(5(3+z)-(8z+9)=-4z\). Substitute \(z = -6\): \(5(3 - 6) - (8(-6) + 9) = -4(-6)\). Simplifying each side gives: \(5(-3) - (-48 + 9) = 24\) becomes \(-15 + 39 = 24\), which simplifies to \(24 = 24\). Both sides are equal, confirming that \(z = -6\) is correct.

Key Concepts

Algebraic Simplification

Simplifying equations is a fundamental concept in algebra that involves transforming an equation into its simplest form. This process can make solving the equation much easier. In the given problem, we started with the expression \(5(3+z) - (8z+9)\). The first step in simplification is to perform all possible arithmetic operations and combine like terms.

• Combine all constant terms together.
• Combine all terms that involve the same variable into a single term.

By distributing the constant 5 across the bracket \((3+z)\), and managing the distribution of the negative sign, we simplify the expression to \(6 - 3z\). By following these simplification steps, we can make equations more manageable.

Distributive Property

The distributive property of multiplication over addition is a key principle that simplifies complex expressions. It states that multiplying a number by a sum is the same as doing each multiplication separately. For example, \(a(b + c) = ab + ac\).In our problem, the distributive property was used with the expression \(5(3+z)\). Applying this property results in \(15 + 5z\). This is because we multiply 5 by each term inside the parentheses separately. Additionally, using the distributive property to distribute the negative sign across \((8z + 9)\) gives us \(-8z - 9\), further simplifying the equation.Understanding the distributive property helps in quickly breaking down expressions into smaller parts, which can then be further simplified.

Isolating Variables

Isolating variables is a critical step in solving algebraic equations. The goal is to have the variable on one side of the equation, with everything else on the opposite side. This allows us to solve for the variable directly. In the example, after simplifying the equation to \(6 - 3z = -4z\), we isolate \(z\) by adding \(3z\) to both sides. This is crucial because it puts all terms containing \(z\) together. Now the equation reads \(6 = -z\). By taking these steps systematically, we effectively work towards solving the equation completely. Let's focus then on the next part, which is finding the exact value of \(z\).

Checking Solutions

Checking solutions ensures that our calculation is correct and verifies that the value found for the variable satisfies the original equation. In this exercise, after obtaining \(z = -6\), we substitute \(-6\) back into the original equation, \(5(3+z)-(8z+9)=-4z\). This step is crucial as it confirms that both sides of the equation are balanced. By calculating separately, we simplify both sides until we determine that both result in \(24 = 24\).

• Checking substitutes the possible solution back into the original equation.
• It helps us ensure there are no computational errors.
• Confirms the validity of the solution with respect to the initial problem.

Always remember, verifying your solution is just as vital as solving the equation itself, as it reassures the accuracy of your work.