Solve For $z$. Express Your Answer As A Proper Or Improper Fraction In Simplest Terms. − 5 6 = − 9 11 Z + 1 2 -\frac{5}{6}=-\frac{9}{11} Z+\frac{1}{2} − 6 5 ​ = − 11 9 ​ Z + 2 1 ​ Answer Z = Z= Z = □ \square □

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Introduction

In algebra, solving for a variable means isolating it on one side of the equation. This is a crucial skill to master, as it allows us to find the value of the variable and make predictions about real-world situations. In this article, we will focus on solving for z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}.

Understanding the Equation

Before we can solve for z, we need to understand the equation. The equation is a linear equation, which means it can be written in the form ax+b=cax + b = c. In this case, the equation is −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}. Our goal is to isolate z, which means we need to get rid of the other terms in the equation.

Isolating the Variable

To isolate z, we need to get rid of the term −911z-\frac{9}{11} z. We can do this by adding 911z\frac{9}{11} z to both sides of the equation. This will cancel out the term −911z-\frac{9}{11} z on the left-hand side of the equation.

-\frac{5}{6} = -\frac{9}{11} z + \frac{1}{2}
\frac{9}{11} z + \frac{5}{6} = \frac{1}{2}

Simplifying the Equation

Now that we have added 911z\frac{9}{11} z to both sides of the equation, we need to simplify the equation. We can do this by finding a common denominator for the fractions on the left-hand side of the equation.

\frac{9}{11} z + \frac{5}{6} = \frac{1}{2}
\frac{54}{66} z + \frac{55}{66} = \frac{33}{66}

Combining Like Terms

Now that we have a common denominator, we can combine the like terms on the left-hand side of the equation. The like terms are the fractions with the same denominator.

\frac{54}{66} z + \frac{55}{66} = \frac{33}{66}
\frac{109}{66} z = \frac{33}{66}

Solving for z

Now that we have combined the like terms, we can solve for z. We can do this by dividing both sides of the equation by 10966\frac{109}{66}.

\frac{109}{66} z = \frac{33}{66}
z = \frac{33}{66} \div \frac{109}{66}
z = \frac{33}{66} \times \frac{66}{109}
z = \frac{33}{109}

Conclusion

In this article, we have solved for z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}. We started by understanding the equation and then isolated the variable by adding 911z\frac{9}{11} z to both sides of the equation. We then simplified the equation by finding a common denominator and combining like terms. Finally, we solved for z by dividing both sides of the equation by 10966\frac{109}{66}. The final answer is 33109\boxed{\frac{33}{109}}.

Frequently Asked Questions

  • What is the value of z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}?
    • The value of z is 33109\frac{33}{109}.
  • How do I isolate the variable in a linear equation?
    • To isolate the variable, you need to get rid of the other terms in the equation. You can do this by adding or subtracting the same value to both sides of the equation.
  • How do I simplify a linear equation?
    • To simplify a linear equation, you need to find a common denominator for the fractions and combine like terms.

Final Answer

The final answer is 33109\boxed{\frac{33}{109}}.

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Introduction

In algebra, solving for a variable means isolating it on one side of the equation. This is a crucial skill to master, as it allows us to find the value of the variable and make predictions about real-world situations. In this article, we will focus on solving for z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}.

Understanding the Equation

Before we can solve for z, we need to understand the equation. The equation is a linear equation, which means it can be written in the form ax+b=cax + b = c. In this case, the equation is −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}. Our goal is to isolate z, which means we need to get rid of the other terms in the equation.

Isolating the Variable

To isolate z, we need to get rid of the term −911z-\frac{9}{11} z. We can do this by adding 911z\frac{9}{11} z to both sides of the equation. This will cancel out the term −911z-\frac{9}{11} z on the left-hand side of the equation.

-\frac{5}{6} = -\frac{9}{11} z + \frac{1}{2}
\frac{9}{11} z + \frac{5}{6} = \frac{1}{2}

Simplifying the Equation

Now that we have added 911z\frac{9}{11} z to both sides of the equation, we need to simplify the equation. We can do this by finding a common denominator for the fractions on the left-hand side of the equation.

\frac{9}{11} z + \frac{5}{6} = \frac{1}{2}
\frac{54}{66} z + \frac{55}{66} = \frac{33}{66}

Combining Like Terms

Now that we have a common denominator, we can combine the like terms on the left-hand side of the equation. The like terms are the fractions with the same denominator.

\frac{54}{66} z + \frac{55}{66} = \frac{33}{66}
\frac{109}{66} z = \frac{33}{66}

Solving for z

Now that we have combined the like terms, we can solve for z. We can do this by dividing both sides of the equation by 10966\frac{109}{66}.

\frac{109}{66} z = \frac{33}{66}
z = \frac{33}{66} \div \frac{109}{66}
z = \frac{33}{66} \times \frac{66}{109}
z = \frac{33}{109}

Conclusion

In this article, we have solved for z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}. We started by understanding the equation and then isolated the variable by adding 911z\frac{9}{11} z to both sides of the equation. We then simplified the equation by finding a common denominator and combining like terms. Finally, we solved for z by dividing both sides of the equation by 10966\frac{109}{66}. The final answer is 33109\boxed{\frac{33}{109}}.

Frequently Asked Questions

Q: What is the value of z in the equation −56=−911z+12-\frac{5}{6}=-\frac{9}{11} z+\frac{1}{2}?

A: The value of z is 33109\frac{33}{109}.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable, you need to get rid of the other terms in the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to find a common denominator for the fractions and combine like terms.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula, which is x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: What is the significance of the variable in an equation?

A: The variable in an equation represents a value that can change, while the constants in the equation represent values that do not change.

Q: How do I determine the value of a variable in an equation?

A: To determine the value of a variable in an equation, you need to isolate the variable by getting rid of the other terms in the equation.

Final Answer

The final answer is 33109\boxed{\frac{33}{109}}.