Problem 8 - Prealgebra

Question

Check to see if the number to the right of each of the following equations is the solution to the equation. $$4 x+5=2 x-1 ;-6$$

Step-by-Step Solution

Answer

The number -6 is not a solution to the equation.

Step 1: Set the equation

First, review the given equation and the number to determine if it is a solution. The equation is \(4x + 5 = 2x - 1\) and the number we need to check is \(-6\).

Step 2: Substitute the value into the equation

Substitute \(-6\) for \(x\) in the equation to check if both sides are equal:\(4(-6) + 5 = 2(-6) - 1\)

Step 3: Simplify the left side

Compute the left side of the equation:\(4(-6) + 5 = -24 + 5 = -19\)

Step 4: Simplify the right side

Compute the right side of the equation:\(2(-6) - 1 = -12 - 1 = -13\)

Step 5: Compare both sides of the equation

Compare the results from Steps 3 and 4. The left side is \(-19\), and the right side is \(-13\). Since \(-19 eq -13\), the number \(-6\) is not a solution of the equation.

Key Concepts

Equation Solving

Solving equations is a fundamental skill in mathematics that allows you to find unknown values. The process involves manipulating the equation to find the value of the variable that makes the equation true. When the equation says \(4x + 5 = 2x - 1\), it means both sides must equal each other when the correct value of \(x\) is used. This goal to balance both sides makes equation solving an engaging puzzle.
To check if a given number is a solution, you substitute it into the equation and see if it makes a true statement.

• If both sides equal, you've found the solution.
• If they don't, the number is not a solution.

This way of checking solutions ensures you understand not just what the answer is, but why it is the answer.
Using these steps reinforces your algebra skills and builds a deeper understanding of mathematical concepts.

Substitution Method in Algebra

The substitution method in algebra is a valuable technique, especially when checking solutions for equations. It involves replacing a variable with a given number to see how it affects the equation.
In the problem we explored, you took the number \(-6\) and substituted it for \(x\) in the equation: \(4x + 5 = 2x - 1\). This requires you to perform the operations step by step just as you would do in arithmetic.
Here's a quick rundown:

• Insert \(-6\) in place of each \(x\).
• Calculate both sides independently.
• Compare the results of each side.

This method not only checks if the number fits but also strengthens your algebraic manipulation skills by requiring precise arithmetic with each substitution.
While it might seem simple, mastering substitution can make more complex algebra problems much easier to handle.

Equation Simplification

Simplifying equations involves reducing them to their simplest form to make it easier to compare both sides. In our example, we substituted \(-6\) for \(x\), leading to: \(4(-6) + 5\) on the left and \(2(-6) - 1\) on the right.
Here’s the simplification in action:

• Calculate: \(4(-6) = -24\)
• Then add: \(-24 + 5 = -19\)
• On the right side: \(2(-6) = -12\)
• Then subtract: \(-12 - 1 = -13\)

The end goal is to reach numbers on both sides that are easy to compare. In this case, you compare \(-19\) to \(-13\). Because they are not equal, \(-6\) is not a solution.
Mastering equation simplification means you can distill problems down to their core, making it easier to solve a variety of mathematical challenges.