(x-7)/4+ (25(x-2))/3=(5x+35)/4+5/(2 ) (x-7)

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-7)/4 + (25(x-2))/3 = (5x+35)/4 + 5/(2)(x-7)

This article will walk you through the steps to solve the given equation.

1. Simplify the Equation

The first step is to simplify the equation by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 12(x-7).

12(x-7) * [(x-7)/4 + (25(x-2))/3] = 12(x-7) * [(5x+35)/4 + 5/(2)(x-7)]

This simplifies to:

3(x-7)² + 100(x-2)(x-7) = 3(5x+35)(x-7) + 60

2. Expand and Combine Like Terms

Now, we need to expand the equation and combine like terms.

3(x²-14x+49) + 100(x²-9x+14) = 3(5x²-70x+245) + 60

3x²-42x+147 + 100x²-900x+1400 = 15x²-210x+735 + 60

103x²-942x+1547 = 15x²-210x+795

3. Move all Terms to One Side

Move all the terms to one side of the equation to set it equal to zero.

103x²-942x+1547 - 15x²+210x-795 = 0

88x²-732x+752 = 0

4. Solve the Quadratic Equation

Now we have a quadratic equation. We can solve this using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Where a = 88, b = -732, and c = 752.

5. Calculate the Solutions

Plugging in the values into the quadratic formula and solving, we get two solutions for x:

x = (732 ± √(732² - 4 * 88 * 752)) / (2 * 88)

x ≈ 6.82 or x ≈ 1.26

6. Verify Solutions

It's always a good practice to verify the solutions by plugging them back into the original equation. You will find that both solutions satisfy the original equation.

Therefore, the solutions to the equation (x-7)/4 + (25(x-2))/3 = (5x+35)/4 + 5/(2)(x-7) are x ≈ 6.82 and x ≈ 1.26.