Solve For { X $}$ In The Equation:${ \frac{4x + 7}{2} = X + 8 }$

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a given equation. In this article, we will focus on solving the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 } to find the value of { x $}$. This equation involves fractions, and we will use algebraic techniques to simplify and solve it.

Understanding the Equation

The given equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }. To solve for { x $}$, we need to isolate the variable on one side of the equation. The first step is to simplify the left-hand side of the equation by multiplying both sides by 2 to eliminate the fraction.

Simplifying the Equation

By multiplying both sides of the equation by 2, we get:

4x+7=2(x+8){ 4x + 7 = 2(x + 8) }

This simplifies to:

4x+7=2x+16{ 4x + 7 = 2x + 16 }

Isolating the Variable

Now that we have simplified the equation, we can isolate the variable { x $}$ by moving all the terms involving { x $}$ to one side of the equation. We can do this by subtracting 2x from both sides of the equation.

Subtracting 2x from Both Sides

Subtracting 2x from both sides of the equation gives us:

4x−2x+7=2x−2x+16{ 4x - 2x + 7 = 2x - 2x + 16 }

This simplifies to:

2x+7=16{ 2x + 7 = 16 }

Solving for { x $}$

Now that we have isolated the variable, we can solve for { x $}$ by subtracting 7 from both sides of the equation.

Subtracting 7 from Both Sides

Subtracting 7 from both sides of the equation gives us:

2x+7−7=16−7{ 2x + 7 - 7 = 16 - 7 }

This simplifies to:

2x=9{ 2x = 9 }

Final Step: Dividing by 2

To find the value of { x $}$, we need to divide both sides of the equation by 2.

Dividing Both Sides by 2

Dividing both sides of the equation by 2 gives us:

2x2=92{ \frac{2x}{2} = \frac{9}{2} }

This simplifies to:

x=92{ x = \frac{9}{2} }

Conclusion

In this article, we solved the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 } to find the value of { x $}$. We simplified the equation by multiplying both sides by 2, isolated the variable by subtracting 2x from both sides, and finally solved for { x $}$ by dividing both sides by 2. The final answer is { x = \frac{9}{2} $}$.

Frequently Asked Questions

Q: What is the value of { x $}$ in the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }?

A: The value of { x $}$ is { \frac{9}{2} $}$.

Q: How do I simplify the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }?

A: To simplify the equation, multiply both sides by 2 to eliminate the fraction.

Q: How do I isolate the variable { x $}$ in the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }?

A: To isolate the variable, subtract 2x from both sides of the equation.

Q: How do I solve for { x $}$ in the equation 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }?

A: To solve for { x $}$, divide both sides of the equation by 2.

Additional Resources

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Introduction

Solving equations with fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will provide answers to some of the most frequently asked questions about solving equations with fractions.

Q&A

Q: What is the first step in solving an equation with fractions?

A: The first step in solving an equation with fractions is to simplify the equation by eliminating the fractions. This can be done by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: How do I simplify an equation with fractions?

A: To simplify an equation with fractions, multiply both sides of the equation by the LCM of the denominators. For example, if the equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }, the LCM of the denominators is 2, so we multiply both sides by 2 to get:

4x+7=2(x+8){ 4x + 7 = 2(x + 8) }

Q: How do I isolate the variable in an equation with fractions?

A: To isolate the variable in an equation with fractions, subtract the term with the variable from both sides of the equation. For example, if the equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }, we can subtract x from both sides to get:

4x+72−x=8{ \frac{4x + 7}{2} - x = 8 }

Q: How do I solve for the variable in an equation with fractions?

A: To solve for the variable in an equation with fractions, divide both sides of the equation by the coefficient of the variable. For example, if the equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }, we can divide both sides by 2 to get:

x=92{ x = \frac{9}{2} }

Q: What is the least common multiple (LCM) of the denominators in an equation with fractions?

A: The LCM of the denominators in an equation with fractions is the smallest number that is a multiple of all the denominators. For example, if the equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }, the LCM of the denominators is 2.

Q: How do I find the LCM of the denominators in an equation with fractions?

A: To find the LCM of the denominators in an equation with fractions, list the multiples of each denominator and find the smallest number that is common to all the lists. For example, if the equation is 4x+72=x+8{ \frac{4x + 7}{2} = x + 8 }, the multiples of 2 are 2, 4, 6, 8, 10, ... and the multiples of 1 are 1, 2, 3, 4, 5, ... The smallest number that is common to both lists is 2, so the LCM of the denominators is 2.

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Conclusion

Solving equations with fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. By simplifying the equation, isolating the variable, and solving for the variable, we can find the solution to the equation. We hope that this article has provided you with the information and resources you need to solve equations with fractions.