Problem 103 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$4(6-z)+7 z=0 ;-8$$

Step-by-Step Solution

Answer

Yes, -8 is a solution to the equation.

Step 1: Substitute -8 for z

Substitute the given number (-8) in place of z in the given equation. It results in: \(4(6- -8)+7(-8)=0\). This simplifies to \(4(6+8)+7(-8)=0\).

Step 2: Simplify

Now, further simplify this equation. \(4(14)-56=0\), which gives -56=-56.

Step 3: Verify

As the simplified equation -56 = -56 is correct, conclude that -8 is indeed a solution to the original equation.

Key Concepts

Substitution Method

The substitution method is a powerful technique often used to solve algebraic equations. By substituting a given value into an equation, you can check to see if it satisfies the equation. In this case, we're given the equation \(4(6-z)+7z=0\) and the number \(-8\). Our goal is to check if \(-8\) is a solution. First, we replace \(z\) in the equation with \(-8\). This gives us:

• First, calculate \(6 - (-8)\), which is equivalent to \(6 + 8\).
• This results in the expression \(4 \times 14 + 7 \times (-8)\).

By using substitution, we transform the original equation into something we can verify. This method is particularly useful because it allows us to test if a particular value works without solving the entire equation from scratch. Simply put, it's about placing the given number into the equation and checking if the result holds true.

Solution Verification

Once you have substituted the value into the equation, it's time for verification. Verification is necessary to confirm that our substitution has been done correctly and the value is indeed a solution.After substituting \(-8\) for \(z\) in the equation \(4(6-z) + 7z = 0\), and simplifying, you obtain \(4(14) - 56\).

• Calculate \(4(14)\) which equals \(56\).
• Then do \(56 - 56\) which simplifies to \(0\).

The left-hand side of the equation equals the right-hand side, which is zero in this case. Since both sides are equal, we can verify that \(-8\) is indeed a solution to the original equation. Verification is an essential step that ensures the accuracy of your solution. If both sides of the equation are equal, you've successfully verified the solution.

Simplification Steps

Simplification is all about making mathematical expressions easier to understand and solve. After the substitution method, simplifying the equation helps in verifying the solution. In our example, simplifying the equation \(4(14) - 56 = 0\) takes a few straightforward steps.Begin by calculating the multiplication:

• Start with \(4 \times 14\), resulting in \(56\).
• Move on to the subtraction part, \(56 - 56\), yielding \(0\).

When simplifying expressions, always handle multiplication and division before addition and subtraction, following the order of operations (PEMDAS/BODMAS).Ultimately, simplification will present the equation in a form that's easy to analyze. By reducing it to a basic equality, like \(-56 = -56\), you are effectively showing that the left side matches the right side. That synonymous nature in the equation confirms that \(-8\) is a true solution.