To solve for \( x \) in the equation
\[ \frac{2}{1}(6x - 10) = \frac{7x + 15}{3x - 5} \]
we can start by simplifying the equation. Since \(\frac{2}{1} = 2\), we can rewrite it as:
\[ 2(6x - 10) = \frac{7x + 15}{3x - 5} \]
Now, distribute the 2 on the left side:
\[ 12x - 20 = \frac{7x + 15}{3x - 5} \]
Next, we need to eliminate the fraction by multiplying both sides by \( (3x - 5) \):
\[ (12x - 20)(3x - 5) = 7x + 15 \]
Expanding the left side, we apply the distributive property:
\[ 12x(3x) - 12x(5) - 20(3x) + 20(5) = 7x + 15 \]
This simplifies to:
\[ 36x^2 - 60x - 60x + 100 = 7x + 15 \]
Combining like terms gives us:
\[ 36x^2 - 120x + 100 = 7x + 15 \]
To set the equation to zero, subtract \(7x\) and \(15\) from both sides:
\[ 36x^2 - 120x - 7x + 100 - 15 = 0 \]
This simplifies to:
\[ 36x^2 - 127x + 85 = 0 \]
Now, let's use the quadratic formula to solve for \(x\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 36\), \(b = -127\), and \(c = 85\). Plugging in these values:
\[ x = \frac{-(-127) \pm \sqrt{(-127)^2 - 4 \cdot 36 \cdot 85}}{2 \cdot 36} \]
Calculating \(b^2\):
\[ (-127)^2 = 16129 \]
Calculating \(4ac\):
\[ 4 \cdot 36 \cdot 85 = 12240 \]
Now, substituting back into the quadratic formula:
\[ x = \frac{127 \pm \sqrt{16129 - 12240}}{72} \] \[ x = \frac{127 \pm \sqrt{3889}}{72} \]
Calculating the square root of \(3889 \approx 62.23\):
Thus, we have:
\[ x = \frac{127 \pm 62.23}{72} \]
Calculating the two possible values:
- \(x = \frac{127 + 62.23}{72} \approx \frac{189.23}{72} \approx 2.63\)
- \(x = \frac{127 - 62.23}{72} \approx \frac{64.77}{72} \approx 0.9\)
Thus, the final values of \( x \) could be approximately \( 2.63 \) or \( 0.9 \). Depending on which value the problem is concerned with, you'd report that value.
If we simplify further and focus on the integer or simple rational value of \( x \), we can conclude. Based on the typical approach in educational contexts, we might select:
\[ x \approx 2.63 \]
However, if you want the simpler value \( x \approx 0.9 \) too might be valid, depending on your specific needs or the problem's context.
Final Answer:
For more precision, please check the problem context. If precise answer needed based on approximations or integer values required from the context or review the whole solving process for validation:
Or simply, carry out pure calculations to exact lead equations to \(\approx 2.63\) or \(\approx 0.9\) based on further iterations or checks.
1 answer