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Chapter 2: Problem 79

Check whether the given number is a solution of the equation. $$6 z+5=8 z-12 ; 8.5$$

Short Answer

Expert verified
Yes, the given number 8.5 is a solution of the equation \(6z + 5 = 8z - 12\).

Step by step solution

01

Understand the Equation

The given equation is \(6z + 5 = 8z - 12\). It is a linear equation in one variable.
02

Substituting the Given Value

Substitute the given number, 8.5, into the equation wherever you find 'z'. So the equation becomes \(6(8.5) + 5 = 8(8.5) - 12\)
03

Simplify Both Sides

Simplify both sides of the equation. You'll get \[51 + 5 = 68 - 12\]. Further simplification gives \[56 = 56\]
04

Check the Equality

The two sides of the equation are equal, hence, 8.5 is a solution of the given equation.

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Key Concepts

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

Solution of Equations
Solving equations is like solving a mystery. Our goal is to find the value of a variable that makes the equation true. In the equation provided, we have a linear equation: \(6z + 5 = 8z - 12\). Linear means it has no exponents other than 1. Each side of the equation represents the same thing; they are two sides of a balance. Our task is to figure out if a given number, in this case, 8.5, is the one that keeps this balance intact. Here's what we do:
  • Think of the equation as a statement that the two expressions are equal.
  • Substituting a number for the variable checks if that number makes the statement true.
  • If both sides are equal after substituting, then that number is a true solution.
If the balancing act succeeds, the number is a solution. Otherwise, it's not.
Substituting Values
Substituting values into an equation is like putting pieces into a puzzle. The given number, 8.5, replaces 'z' everywhere in the equation. Substitution helps us test if this number is the right piece for the puzzle, ensuring the equation stays balanced.Let's step through the substitution:
  • Start with the original equation: \(6z + 5 = 8z - 12\).
  • When substituting, replace every instance of \(z\) with 8.5: \(6(8.5) + 5 = 8(8.5) - 12\).
This replacement transforms the abstract into more concrete numbers, making it easier to work with. It helps us focus on calculation next, leading us to more clarity about whether our number is correct. Testing numbers this way ensures we find the true solution effectively.
Simplifying Equations
Simplifying equations involves streamlining our calculations by doing arithmetic on both sides until we simplify them down to basic results. It makes things far easier to check and ensures clarity.Here's how we simplify:
  • Calculate the expression on each side of the equation: For \(6(8.5) + 5\) and \(8(8.5) - 12\).
  • Compute the left side: \(51 + 5 = 56\).
  • Then compute the right side: \(68 - 12 = 56\).
Finding that both sides result in the same number confirms our substitution worked perfectly. If they didn't match, it would mean the number we substituted is not a solution. Simplifying isn't just math鈥攊t鈥檚 clarity at work, making sure everything aligns perfectly.

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Most popular questions from this chapter

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -4(t-8) $$

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