Deepak's Solution$[ \begin{tabular}{|c|l|c|} \hline & \multicolumn{1}{|c|}{Steps} & Resulting Equations \ \hline 1 & Use The Distributive Property To Simplify. & 5 2 − 5 + 4 X − 3 X = 7 4 \frac{5}{2} - 5 + 4x - 3x = \frac{7}{4} 2 5 ​ − 5 + 4 X − 3 X = 4 7 ​ \ \hline 2 & Simplify By Combining

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Problem Description

Deepak is a high school student who is struggling with a math problem. He has been given an equation to solve, but he is not sure how to approach it. The equation is:

52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}

Deepak wants to simplify the equation and solve for x. He has been trying to use the distributive property, but he is not sure if he is doing it correctly.

Step 1: Use the Distributive Property to Simplify

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. In this case, we can use the distributive property to simplify the equation.

The distributive property states that for any real numbers a, b, and c:

a(b+c)=ab+aca(b + c) = ab + ac

We can apply this property to the equation by multiplying the 4x and -3x terms by the 5/2 term.

52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}

Using the distributive property, we can rewrite the equation as:

52−5+4x−3x=52⋅4x−52⋅3x\frac{5}{2} - 5 + 4x - 3x = \frac{5}{2} \cdot 4x - \frac{5}{2} \cdot 3x

Simplifying the equation, we get:

52−5+4x−3x=10x−152x\frac{5}{2} - 5 + 4x - 3x = 10x - \frac{15}{2}x

Step 2: Simplify by Combining Like Terms

Now that we have simplified the equation using the distributive property, we can combine like terms to make it easier to solve.

Like terms are terms that have the same variable raised to the same power.

In this case, we have two like terms: 4x and -3x. We can combine these terms by adding their coefficients.

4x−3x=(4−3)x=x4x - 3x = (4 - 3)x = x

So, the simplified equation is:

52−5+x=74\frac{5}{2} - 5 + x = \frac{7}{4}

Step 3: Add 5 to Both Sides of the Equation

To isolate the variable x, we need to get rid of the constant term on the left-hand side of the equation. We can do this by adding 5 to both sides of the equation.

52−5+x=74\frac{5}{2} - 5 + x = \frac{7}{4}

Adding 5 to both sides, we get:

52+x=74+5\frac{5}{2} + x = \frac{7}{4} + 5

Step 4: Simplify the Right-Hand Side of the Equation

Now that we have added 5 to both sides of the equation, we can simplify the right-hand side.

To add fractions, we need to have a common denominator.

In this case, the common denominator is 4. So, we can rewrite the equation as:

52+x=74+204\frac{5}{2} + x = \frac{7}{4} + \frac{20}{4}

Simplifying the right-hand side, we get:

52+x=274\frac{5}{2} + x = \frac{27}{4}

Step 5: Multiply Both Sides of the Equation by 2

To get rid of the fraction on the left-hand side of the equation, we can multiply both sides by 2.

52+x=274\frac{5}{2} + x = \frac{27}{4}

Multiplying both sides by 2, we get:

5+2x=2725 + 2x = \frac{27}{2}

Step 6: Subtract 5 from Both Sides of the Equation

To isolate the variable x, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting 5 from both sides of the equation.

5+2x=2725 + 2x = \frac{27}{2}

Subtracting 5 from both sides, we get:

2x=272−52x = \frac{27}{2} - 5

Step 7: Simplify the Right-Hand Side of the Equation

Now that we have subtracted 5 from both sides of the equation, we can simplify the right-hand side.

To subtract fractions, we need to have a common denominator.

In this case, the common denominator is 2. So, we can rewrite the equation as:

2x=272−1022x = \frac{27}{2} - \frac{10}{2}

Simplifying the right-hand side, we get:

2x=1722x = \frac{17}{2}

Step 8: Divide Both Sides of the Equation by 2

To solve for x, we need to get rid of the coefficient on the left-hand side of the equation. We can do this by dividing both sides by 2.

2x=1722x = \frac{17}{2}

Dividing both sides by 2, we get:

x=174x = \frac{17}{4}

The final answer is 174\boxed{\frac{17}{4}}.

Conclusion

In this article, we have shown how to use the distributive property to simplify an equation and solve for x. We have also shown how to combine like terms, add and subtract fractions, and divide both sides of the equation by a coefficient. By following these steps, we can solve a wide range of algebraic equations.

Discussion

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It is a powerful tool that can be used to simplify complex equations and solve for variables.

In this article, we have shown how to use the distributive property to simplify an equation and solve for x. We have also shown how to combine like terms, add and subtract fractions, and divide both sides of the equation by a coefficient.

The distributive property is a key concept in algebra that can be used to solve a wide range of problems. It is a fundamental concept that is used in many areas of mathematics, including calculus, linear algebra, and differential equations.

Final Answer

The final answer is 174\boxed{\frac{17}{4}}.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about Deepak's solution to the equation 52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any real numbers a, b, and c:

a(b+c)=ab+aca(b + c) = ab + ac

Q: How do I use the distributive property to simplify an equation?

A: To use the distributive property to simplify an equation, you need to multiply the single term by each of the multiple terms. For example, in the equation 52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}, we can use the distributive property to simplify the equation by multiplying the 4x and -3x terms by the 52\frac{5}{2} term.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable. For example, in the equation 4x−3x=x4x - 3x = x, the terms 4x and -3x are like terms because they have the same variable (x) raised to the same power (1).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, in the equation 4x−3x=x4x - 3x = x, we can combine the like terms by adding their coefficients: 4−3=14 - 3 = 1. So, the simplified equation is xx.

Q: What is the difference between adding and subtracting fractions?

A: When adding fractions, you need to have a common denominator. When subtracting fractions, you also need to have a common denominator. For example, in the equation 52+x=74+5\frac{5}{2} + x = \frac{7}{4} + 5, we can add the fractions by finding a common denominator (4) and then adding the numerators: 52⋅22+x=104+x=104+44x\frac{5}{2} \cdot \frac{2}{2} + x = \frac{10}{4} + x = \frac{10}{4} + \frac{4}{4}x.

Q: How do I solve for x in an equation?

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

Q: What is the final answer to the equation 52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}?

A: The final answer to the equation 52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4} is 174\boxed{\frac{17}{4}}.

Conclusion

In this article, we have answered some of the most frequently asked questions about Deepak's solution to the equation 52−5+4x−3x=74\frac{5}{2} - 5 + 4x - 3x = \frac{7}{4}. We have also provided a step-by-step guide on how to use the distributive property to simplify an equation and solve for x.

Final Answer

The final answer is 174\boxed{\frac{17}{4}}.