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Chapter 2: Problem 45

Solve equation, and check your solution. \(\frac{z}{6}=12\)

Short Answer

Expert verified
z = 72

Step by step solution

01

Understand the equation

The given equation is \(\frac{z}{6} = 12\). This means that when z is divided by 6, the result is 12.
02

Isolate the variable

To solve for z, multiply both sides of the equation by 6 to get rid of the fraction. This yields \[z = 12 \times 6.\]
03

Calculate the value

Perform the multiplication: \[z = 72.\]
04

Check the solution

Substitute z = 72 back into the original equation to verify the solution: \(\frac{72}{6} = 12\). Since the equation holds true, the solution is correct.

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Key Concepts

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

Isolating Variables
One of the core steps in solving algebraic equations is isolating the variable. This means we want to get our unknown variable, here represented as z, all by itself on one side of the equation.
In our example equation \(\frac{z}{6} = 12\), z is currently divided by 6.
To isolate z, we need to do the opposite operation. Division by 6 can be reversed by multiplication.
  • Multiply both sides of the equation by 6:
This is how you remove the fraction and isolate the variable.
This transforms \(\frac{z}{6} = 12\) into z = 12 * 6.

So, always remember the goal of isolating variables: get the unknown alone. By performing the right operations on both sides of the equation, you can steadily work towards finding the value of the variable.
Checking Solutions
After isolating the variable and finding a solution, it's crucial to check that your solution is correct.
In our example, we found that z = 72.
To verify:
  • Substitute z back into the original equation:
Originally, we had \(\frac{z}{6} = 12\)
Now, replace z with 72:
  • \(\frac{72}{6} = 12\)
This simplifies to 12 = 12, which confirms our solution is correct.

Checking solutions helps avoid mistakes.
It ensures the work you've done to isolate the variable and solve the equation is accurate.

Always make checking your solution a habit in algebra!
Multiplication in Algebra
Multiplication plays a key role in algebra, especially when isolating variables and solving equations.
For instance, in our example \(\frac{z}{6} = 12\), multiplication was used to nullify the division by 6.
By multiplying both sides by 6, we simplified the equation to z = 72.
  • Use multiplication to flip division:
  • Example: \(\frac{z}{6} = 12\ \rightarrow z = 12 \times 6 = 72\)
  • Use multiplication to balance equations:
  • If both sides are multiplied by the same number, the equality holds true.
Multiplication helps re-arranging and simplifying equations.
If applied correctly, it can make complicated equations look much simpler.
So, mastering multiplication in algebra will empower you to tackle a variety of problems effectively.

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Most popular questions from this chapter

Convert each decimal to a percent. $$ 0.983 $$

Find the value of \(y\) when \((a) x=-2\) and \((b) x=4 .\) Sections 1.3 and 2.3 $$ y=4-3 x $$

Translate each statement into an inequality. Use \(x\) as the variable. You must be at least 18 yr old to vote.

Write each phrase as a mathematical expression using x as the variable. See Sections \(1.3,1.5\) \(1.6,\) and 1.8 The quotient of \(-6\) and a nonzero number

Write each phrase as a mathematical expression using x as the variable. See Sections \(1.3,1.5\) \(1.6,\) and 1.8 A number decreased by 9
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