Which Equations Have The Same Value Of $x$ As 5 6 X + 2 3 = − 9 \frac{5}{6} X+\frac{2}{3}=-9 6 5 ​ X + 3 2 ​ = − 9 ? Select Three Options.A. 6 ( 5 6 X + 2 3 ) = − 9 6\left(\frac{5}{6} X+\frac{2}{3}\right)=-9 6 ( 6 5 ​ X + 3 2 ​ ) = − 9 B. 6 ( 5 6 X + 2 3 ) = − 9 ( 6 6\left(\frac{5}{6} X+\frac{2}{3}\right)=-9(6 6 ( 6 5 ​ X + 3 2 ​ ) = − 9 ( 6 ]C. $5

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the process of solving linear equations, with a focus on the equation $\frac{5}{6} x+\frac{2}{3}=-9$. We will also examine three options that claim to have the same value of $x$ as the given equation and determine which ones are correct.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. The general form of a linear equation is $ax + b = c$, where $a$, $b$, and $c$ are constants. To solve a linear equation, we need to isolate the variable $x$ on one side of the equation.

Solving the Given Equation

The given equation is $\frac{5}{6} x+\frac{2}{3}=-9$. To solve for $x$, we need to isolate the term with $x$ on one side of the equation. We can start by subtracting $\frac{2}{3}$ from both sides of the equation:

$$\frac{5}{6} x+\frac{2}{3}-\frac{2}{3}=-9-\frac{2}{3}$$

This simplifies to:

$$\frac{5}{6} x=-9-\frac{2}{3}$$

To simplify the right-hand side of the equation, we can find a common denominator, which is 6:

$$\frac{5}{6} x=-\frac{27}{6}-\frac{4}{6}$$

Combining the fractions on the right-hand side, we get:

$$\frac{5}{6} x=-\frac{31}{6}$$

To isolate $x$, we can multiply both sides of the equation by the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$:

$$x=-\frac{31}{6}\cdot\frac{6}{5}$$

This simplifies to:

$$x=-\frac{31}{5}$$

Evaluating the Options

Now that we have solved the given equation, we can evaluate the three options that claim to have the same value of $x$.

Option A: $6\left(\frac{5}{6} x+\frac{2}{3}\right)=-9$

To evaluate this option, we can start by distributing the 6 to the terms inside the parentheses:

$$6\left(\frac{5}{6} x\right)+6\left(\frac{2}{3}\right)=-9$$

This simplifies to:

$$5x+4=-9$$

Subtracting 4 from both sides of the equation, we get:

$$5x=-9-4$$

This simplifies to:

$$5x=-13$$

Dividing both sides of the equation by 5, we get:

$$x=-\frac{13}{5}$$

This is not equal to the value of $x$ we found in the previous section, so option A is incorrect.

Option B: $6\left(\frac{5}{6} x+\frac{2}{3}\right)=-9(6)$

To evaluate this option, we can start by distributing the 6 to the terms inside the parentheses:

$$6\left(\frac{5}{6} x\right)+6\left(\frac{2}{3}\right)=-9(6)$$

This simplifies to:

$$5x+4=-54$$

Subtracting 4 from both sides of the equation, we get:

$$5x=-54-4$$

This simplifies to:

$$5x=-58$$

Dividing both sides of the equation by 5, we get:

$$x=-\frac{58}{5}$$

This is not equal to the value of $x$ we found in the previous section, so option B is incorrect.

Option C: $5x+4=-9$

To evaluate this option, we can start by subtracting 4 from both sides of the equation:

$$5x=-9-4$$

This simplifies to:

$$5x=-13$$

Dividing both sides of the equation by 5, we get:

$$x=-\frac{13}{5}$$

This is equal to the value of $x$ we found in the previous section, so option C is correct.

Conclusion

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. The general form of a linear equation is $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable $x$ on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to simplify the equation by combining like terms and eliminating any fractions.

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, you can multiply both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the LCM of a set of numbers?

A: The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Q: How do I find the LCM of a set of numbers?

A: To find the LCM of a set of numbers, you can list the multiples of each number and find the smallest number that appears in all of the lists.

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 (in this case, $x$) 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 can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula and simplify.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation, while a system of linear equations is a set of two or more equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the method of substitution or the method of elimination.

Q: What is the method of substitution?

A: The method of substitution is a method of solving a system of linear equations by substituting one equation into the other equation.

Q: What is the method of elimination?

A: The method of elimination is a method of solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I choose between the method of substitution and the method of elimination?

A: You can choose between the method of substitution and the method of elimination based on the form of the equations and the variables involved.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving linear equations. We have covered topics such as the definition of a linear equation, the steps involved in solving a linear equation, and the difference between a linear equation and a quadratic equation. We have also discussed the quadratic formula and how to use it to solve quadratic equations. Finally, we have covered the difference between a linear equation and a system of linear equations, and how to solve a system of linear equations using the method of substitution or the method of elimination.