Karen Found That The Solution To $x - 7 + 5x = 36$ Is $x = 6$. Which Of These Could Be The Way She Found The Solution?A. Add $-7$ And $5x$, Then Subtract $x$ From Both Sides Of The Equation.B. Add $x - 7

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 steps involved in solving linear equations, using the example of the equation $x - 7 + 5x = 36$. We will examine the possible methods that Karen, the student, could have used to find the solution $x = 6$.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. Linear equations can be written in the form $ax + b = c$, where $a$, $b$, and $c$ are constants. The goal of solving a linear equation is to isolate the variable, $x$, on one side of the equation.

The Given Equation

The given equation is $x - 7 + 5x = 36$. To solve this equation, we need to isolate the variable $x$ on one side of the equation.

Method A: Adding $-7$ and $5x$

One possible method that Karen could have used to solve the equation is to add $-7$ and $5x$ to both sides of the equation. This would result in the equation:

$x - 7 + 5x + 7 + 5x = 36 + 7$

Simplifying the equation, we get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method B: Adding $x - 7$

Another possible method that Karen could have used to solve the equation is to add $x - 7$ to both sides of the equation. This would result in the equation:

$x - 7 + 5x + x - 7 = 36 + x - 7$

Simplifying the equation, we get:

$7x - 14 = 29 + x - 7$

Now, we can add 14 to both sides of the equation to get:

$7x = 43 + x$

Subtracting $x$ from both sides of the equation, we get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method C: Subtracting $x$ from Both Sides

A more correct method that Karen could have used to solve the equation is to subtract $x$ from both sides of the equation. This would result in the equation:

$x - 7 + 5x - x = 36 - x$

Simplifying the equation, we get:

$4x - 7 = 36 - x$

Now, we can add $x$ to both sides of the equation to get:

$5x - 7 = 36$

Adding 7 to both sides of the equation, we get:

$5x = 43$

Now, we can divide both sides of the equation by 5 to isolate $x$:

$x = \frac{43}{5}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method D: Combining Like Terms

A more correct method that Karen could have used to solve the equation is to combine like terms on the left-hand side of the equation. This would result in the equation:

$x - 7 + 5x = 36$

Combining the like terms $x$ and $5x$, we get:

$6x - 7 = 36$

Now, we can add 7 to both sides of the equation to get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method E: Isolating the Variable

A more correct method that Karen could have used to solve the equation is to isolate the variable $x$ on one side of the equation. This would result in the equation:

$x - 7 + 5x = 36$

Subtracting $5x$ from both sides of the equation, we get:

$x - 7 = 36 - 5x$

Now, we can add $5x$ to both sides of the equation to get:

$x = 36 - 7$

Now, we can simplify the equation to get:

$x = 29$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method F: Using the Distributive Property

A more correct method that Karen could have used to solve the equation is to use the distributive property to expand the left-hand side of the equation. This would result in the equation:

$x - 7 + 5x = 36$

Using the distributive property, we get:

$x + 5x - 7 = 36$

Combining the like terms $x$ and $5x$, we get:

$6x - 7 = 36$

Now, we can add 7 to both sides of the equation to get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Method G: Using Inverse Operations

A more correct method that Karen could have used to solve the equation is to use inverse operations to isolate the variable $x$ on one side of the equation. This would result in the equation:

$x - 7 + 5x = 36$

Subtracting $5x$ from both sides of the equation, we get:

$x - 7 = 36 - 5x$

Now, we can add $5x$ to both sides of the equation to get:

$x = 36 - 7$

Now, we can simplify the equation to get:

$x = 29$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

Conclusion

In conclusion, the correct method that Karen could have used to solve the equation $x - 7 + 5x = 36$ is to combine like terms on the left-hand side of the equation, then isolate the variable $x$ on one side of the equation. This would result in the equation:

$x - 7 + 5x = 36$

Combining the like terms $x$ and $5x$, we get:

$6x - 7 = 36$

Now, we can add 7 to both sides of the equation to get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

However, if we use the correct method, we get:

$x - 7 + 5x = 36$

Subtracting $x$ from both sides of the equation, we get:

$4x - 7 = 36 - x$

Now, we can add $x$ to both sides of the equation to get:

$5x - 7 = 36$

Adding 7 to both sides of the equation, we get:

$5x = 43$

Now, we can divide both sides of the equation by 5 to isolate $x$:

$x = \frac{43}{5}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

However, if we use the correct method, we get:

$x - 7 + 5x = 36$

Combining the like terms $x$ and $5x$, we get:

$6x - 7 = 36$

Now, we can add 7 to both sides of the equation to get:

$6x = 43$

Now, we can divide both sides of the equation by 6 to isolate $x$:

$x = \frac{43}{6}$

This is not the solution that Karen found, which is $x = 6$. Therefore, this method is not the correct way to solve the equation.

However, if we use the correct method, we get:

$x - 7 + 5x = 36$

Subtracting $5x$ from both sides of the equation, we get:

$x - 7 = 36 - 5x$

Now, we can add $5x$ to both sides of the equation to get:

$x = 36 - 7$

Now, we can simplify the equation to get:

$x = 29$

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 steps involved in solving linear equations, using the example of the equation $x - 7 + 5x = 36$. We will also answer some common questions that students may have when solving linear equations.

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. Linear equations can be written in the form $ax + b = c$, 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 on one side of the equation. You can do this by using inverse operations, such as addition and subtraction, to get rid of the constants on the other side of the equation.

Q: What are some common methods for solving linear equations?

A: Some common methods for solving linear equations include:

• Combining like terms on the left-hand side of the equation
• Using inverse operations to isolate the variable on one side of the equation
• Using the distributive property to expand the left-hand side of the equation
• Using inverse operations to get rid of the constants on the other side of the equation

Q: How do I combine like terms on the left-hand side of the equation?

A: To combine like terms on the left-hand side of the equation, you need to add or subtract the coefficients of the like terms. For example, if you have the equation $2x + 3x = 5$, you can combine the like terms by adding the coefficients:

$2x + 3x = 5x$

Q: How do I use inverse operations to isolate the variable on one side of the equation?

A: To use inverse operations to isolate the variable on one side of the equation, you need to add or subtract the same value to both sides of the equation. For example, if you have the equation $x + 2 = 5$, you can use inverse operations to isolate the variable by subtracting 2 from both sides of the equation:

$x + 2 - 2 = 5 - 2$

$x = 3$

Q: How do I use the distributive property to expand the left-hand side of the equation?

A: To use the distributive property to expand the left-hand side of the equation, you need to multiply the coefficient of the variable by the other term in the equation. For example, if you have the equation $2(x + 3) = 5$, you can use the distributive property to expand the left-hand side of the equation:

$2x + 6 = 5$

Q: How do I use inverse operations to get rid of the constants on the other side of the equation?

A: To use inverse operations to get rid of the constants on the other side of the equation, you need to add or subtract the same value to both sides of the equation. For example, if you have the equation $x + 2 = 5$, you can use inverse operations to get rid of the constants by subtracting 2 from both sides of the equation:

$x + 2 - 2 = 5 - 2$

$x = 3$

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Not combining like terms on the left-hand side of the equation
• Not using inverse operations to isolate the variable on one side of the equation
• Not using the distributive property to expand the left-hand side of the equation
• Not using inverse operations to get rid of the constants on the other side of the equation

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By using the methods outlined in this article, you can solve linear equations with ease. Remember to combine like terms on the left-hand side of the equation, use inverse operations to isolate the variable on one side of the equation, use the distributive property to expand the left-hand side of the equation, and use inverse operations to get rid of the constants on the other side of the equation. With practice and patience, you will become a pro at solving linear equations in no time!