Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$4 x=2 x-10 ;-5$$

Step-by-Step Solution

Answer

Yes, -5 is a solution to the equation \(4x = 2x - 10\).

Step 1: Understand the problem

We are given an equation \(4x = 2x - 10\) and a number \(-5\). We are tasked with determining if \(-5\) is a solution to the equation.

Step 2: Substitute the number into the equation

We substitute \(-5\) in the equation wherever \(x\) appears. The equation becomes: \(4(-5) = 2(-5) - 10\)

Step 3: Simplify both sides of the equation

On the left side of the equation, \(4(-5)\) gives \(-20\). On the right side, \(2(-5) - 10\) simplifies to \(-10 - 10\) which equals \(-20\)

Step 4: Compare both sides of the equation

The left side of the equation is \(-20\) and the right side of the equation is also \(-20\). Since the left side equals the right side, the statement is true.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. They involve variables, constants, and arithmetic operations such as addition, subtraction, multiplication, and division.
For example, in the context of our exercise, the equation is given as \(4x = 2x - 10\). Here, \(x\) is the variable and the numbers are constants.
Algebraic equations can vary from very simple to very complex; however, many can be solved using just a few straightforward techniques.

• The ultimate goal is to isolate the variable on one side of the equation, typically the left side.
• Achieving this makes it easier to find what value the variable represents, if any.
• Solutions are numbers that, when substituted into the equation, make the equation true.

Understanding algebraic equations is crucial as they form the foundation for many other concepts in mathematics, including inequalities, functions, and calculus.

Solving Linear Equations

Linear equations are a type of algebraic equation where the highest power of the variable is one.
Such equations take the general form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable.
In our exercise example, the equation is \(4x = 2x - 10\), and we are asked to determine if \(-5\) is a solution.

• Solving these equations often involves the same steps, starting with simplifying both sides.
• Then, we involve inverse operations to isolate the variable.
• Inverse operations refer to operations that undo each other, like addition and subtraction or multiplication and division.

Linear equations can have one unique solution, infinitely many solutions, or no solutions at all, depending on the circumstances.
In our exercise, substituting \(-5\) for \(x\) leads to both sides of the equation being equal, confirming that \(-5\) is indeed a solution.

Substitution Method

The substitution method is an approach often used to solve systems of equations but is also applicable for checking solutions for single equations.
In the context of our example, we use substitution to replace the variable \(x\) with the given number, \(-5\).
The equation changes from \(4x = 2x - 10\) to \(4(-5) = 2(-5) - 10\).

• This method helps determine if the chosen value satisfies the equation.
• If both sides simplify to the same value, the number works as a solution.
• If they do not match, the number is not a solution.

This method provides a concrete way to verify solutions, making it an essential tool for solving not only simple equations but more complex problems too.