Problem 99 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$-7 y+18=-10 y+6 ;-4$$

Step-by-Step Solution

Answer

Yes, -4 is a solution of the given equation.

Step 1: Substitution

Insert the given number, -4, into the equation. The equation \(-7y + 18 = -10y + 6\) turns to \(-7*-4 + 18 = -10*-4 + 6\).

Step 2: Simplification

Simplify both sides of the equation. The left side simplifies to \(28 + 18 = 46\) and the right side simplifies to \(40 + 6 = 46\).

Step 3: Verification

Check if both sides of the equation are equal. Here, both sides are equal (46 = 46), which means that the number -4 is indeed a solution to the equation.

Key Concepts

Equation solving

Solving equations is a fundamental aspect of algebra that involves finding the value of a variable that makes the equation true. An equation is a mathematical statement that asserts the equality of two expressions. In its simplest form, solving an equation means finding the unknown value, often represented by a variable like \(y\), that satisfies the equation.
The process usually involves manipulating the equation to isolate the variable on one side. This might include operations like addition, subtraction, multiplication, and division. A well-solved equation is balanced, meaning what you do to one side you must do to the other.
Remember, the goal is to have the variable on one side and a number on the other. But, before jumping into complex equations, it's crucial to understand the simpler steps and build a strong algebraic foundation.

Substitution method

The substitution method is a technique used in solving equations where you replace the variable with a given number to check if it satisfies the equation.
In the given exercise, the number -4 is substituted into the original equation, \(-7y + 18 = -10y + 6\). Through substitution, the variable \(y\) is replaced: \(-7(-4) + 18\) and \(-10(-4) + 6\).

• This replaces the variable \(y\) with the number \(-4\), allowing you to calculate and simplify each side of the equation while checking for equality.
• This method is beneficial as it checks the validity of a potential solution by direct computation.

This approach is particularly useful when you are given a specific value and need to quickly verify if it is a solution to the equation.

Verification step

The verification step is crucial to confirm the validity of a solution. After substituting the number into the equation and simplifying, it's important to check if both sides of the equation are equal.
In the example problem, after substituting \( -4 \) and simplifying both sides, the left side becomes \(28 + 18 = 46\) and the right side results in \(40 + 6 = 46\). Since both sides of the equation equal 46, the verification is successful, confirming \(-4\) is indeed the correct solution.

• Verification ensures no mistakes were made during substitution or simplification steps.
• It provides a thorough check to confirm the number fits the equation correctly.

This step builds confidence in the solution's accuracy and helps reinforce the understanding of solving equations.