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Chapter 1: Problem 50

Decide whether the given number is a solution of the given equation. $$ \text { Is } 7 \text { a solution of } 3 n-5=10 ? $$

Short Answer

Expert verified
No, 7 is not a solution of the equation.

Step by step solution

01

Substitute the value

First, substitute the number in question, which is 7, into the equation. Replace the variable \( n \) with 7, so the equation becomes \( 3(7) - 5 = 10 \).
02

Perform the multiplication

Multiply the coefficient 3 by 7, resulting in \( 21 \). So the equation is now \( 21 - 5 = 10 \).
03

Perform the subtraction

Subtract 5 from 21, resulting in \( 16 \). Thus, the left side of the equation is now \( 16 \).
04

Compare each side of the equation

Compare the result from the left side (16) with the right side of the original equation (10). Since they are not equal (\( 16 eq 10 \)), the number 7 is not a solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation solving
When it comes to solving equations, understand that an equation is like a balance scale. Each side of the equation represents a different amount of weight, and your goal is to keep the balance. Understanding Equations:
  • An equation is a mathematical statement showing that two expressions are equal.
  • The expressions can be made up of numbers, variables, and operations like addition, subtraction, multiplication, and division.
  • Solving an equation entails finding a value for the variable that makes the equation true, keeping both sides equal.
Think of the equation like a puzzle, where the variable is the missing piece you need to find. In this context, by testing if 7 is a solution to the equation, you are determining if 7 balances the scale of the equation.
Substitution method
The substitution method is a straightforward technique you use to test solutions or solve systems of equations.Steps to Substitution:
  • Initially, replace the variable with the number you wish to test.
  • Perform the necessary mathematical operations as specified in the expression.
  • Evaluate whether the left side of the equation equals the right side.
In our example, we substituted 7 for the variable \( n \) in the equation \( 3n-5=10 \). We then followed through with the calculations: multiplying, subtracting, and finally checking if both sides matched. It's a method often used for checking potential solutions by predicting and testing.
Linear equations
Linear equations are the simplest type of equations you will encounter and are fundamental in algebra.Defining Characteristics of Linear Equations:
  • They have no exponents or powers higher than one.
  • The equation forms a straight line when graphed on a coordinate plane.
  • Usually composed of one variable eliminated through algebraic manipulation.
The equation given \(3n-5=10\) is a clear example of a linear equation. It's all about keeping expressions tidy: multiply, add or subtract, and balance the equation to find what value, if any, satisfies the equation. In this particular exercise, after performing the substitution and calculations, it was determined that 7 does not satisfy the equation, thus revealing that it isn't a solution.

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