Question

question
solve for all values of x.
\\(\frac{x - 7}{x - 4} = \frac{5}{x}\\)
answer attempt 1 out of 2
additional solution no solution
x =
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Explanation:

Step1: Cross - multiply the equation

To solve the equation \(\frac{x - 7}{x - 4}=\frac{5}{x}\), we cross - multiply. Cross - multiplying gives us \(x(x - 7)=5(x - 4)\).

Step2: Expand both sides

Expanding the left - hand side: \(x(x - 7)=x^{2}-7x\).
Expanding the right - hand side: \(5(x - 4)=5x-20\).
So the equation becomes \(x^{2}-7x = 5x-20\).

Step3: Rearrange into standard quadratic form

Subtract \(5x\) from both sides and add 20 to both sides to get the quadratic equation in standard form \(ax^{2}+bx + c = 0\).
\(x^{2}-7x-5x + 20=0\), which simplifies to \(x^{2}-12x + 20 = 0\).

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \(20\) and add up to \(- 12\). The numbers are \(-10\) and \(-2\).
So, \(x^{2}-12x + 20=(x - 10)(x - 2)=0\).

Step5: Solve for x

Set each factor equal to zero:

• If \(x - 10=0\), then \(x = 10\).
• If \(x - 2=0\), then \(x = 2\).

Step6: Check for extraneous solutions

We need to check if these solutions make the original denominators zero.
For the original equation \(\frac{x - 7}{x - 4}=\frac{5}{x}\), the denominators are \(x-4\) and \(x\).

• When \(x = 10\): \(x-4=10 - 4=6

eq0\) and \(x = 10
eq0\). So \(x = 10\) is a valid solution.

• When \(x = 2\): \(x-4=2 - 4=-2

eq0\) and \(x = 2
eq0\). So \(x = 2\) is a valid solution.

Answer:

\(x = 2\) or \(x = 10\)