(x-5)/(x)=(x-6)/(x-3)

Set up the equation: Set up the equation. The given equation is (x-5)/(x) = (x-6)/(x-3). We need to find the value of x that satisfies this equation.Cross-multiply to eliminate: Cross-multiply to eliminate the fractions. (x - 5)(x - 3) = (x - 6)(x) This will give us a quadratic equation to solve for x.Distribute both sides: Distribute both sides of the equation. On the left side: (x - 5)(x - 3) = x^2 - 3x - 5x + 15 On the right side: (x - 6)(x) = x^2 - 6x Now we have x^2 - 3x - 5x + 15 = x^2 - 6x.Simplify both sides: Simplify both sides of the equation. Combine like terms on the left side: x^2 - 8x + 15 The right side remains the same: x^2 - 6x Now we have x^2 - 8x + 15 = x^2 - 6x.Subtract x^2: Subtract x^2 from both sides to eliminate the quadratic term. -8x + 15 = -6x Now we have a linear equation to solve for x.Add 6x to isolate: Add 6x to both sides to isolate the x term on one side. -8x + 6x + 15 = -6x + 6x This simplifies to -2x + 15 = 0.Subtract 15 to solve: Subtract 15 from both sides to solve for x. -2x + 15 - 15 = 0 - 15 This simplifies to -2x = -15.Divide both sides: Divide both sides by -2 to solve for x. -2x / -2 = -15 / -2 This simplifies to x = 7.5.

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Math Problems

Algebra 2

Add, subtract, multiply, and divide polynomials

Full solution

Q. Find `x`.

\frac{x-5}{x}=\frac{x-6}{x-3}

Set up the equation: Set up the equation. $\newline$ The given equation is $(x-5)/(x) = (x-6)/(x-3)$ . $\newline$ We need to find the value of $x$ that satisfies this equation.
Cross-multiply to eliminate: Cross-multiply to eliminate the fractions. $\newline$ $(x - 5)(x - 3) = (x - 6)(x)$ $\newline$ This will give us a quadratic equation to solve for $x$ .
Distribute both sides: Distribute both sides of the equation. $\newline$ On the left side: $(x - 5)(x - 3) = x^2 - 3x - 5x + 15$ $\newline$ On the right side: $(x - 6)(x) = x^2 - 6x$ $\newline$ Now we have $x^2 - 3x - 5x + 15 = x^2 - 6x$ .
Simplify both sides: Simplify both sides of the equation. $\newline$ Combine like terms on the left side: $x^2 - 8x + 15$ $\newline$ The right side remains the same: $x^2 - 6x$ $\newline$ Now we have $x^2 - 8x + 15 = x^2 - 6x$ .
Subtract $x^2$ : Subtract $x^2$ from both sides to eliminate the quadratic term. $\newline$ $-8x + 15 = -6x$ $\newline$ Now we have a linear equation to solve for $x$ .
Add $6x$ to isolate: Add $6x$ to both sides to isolate the $x$ term on one side. $\newline$ $-8x + 6x + 15 = -6x + 6x$ $\newline$ This simplifies to $-2x + 15 = 0$ .
Subtract $15$ to solve: Subtract $15$ from both sides to solve for $x$ . $-2x + 15 - 15 = 0 - 15$ This simplifies to $-2x = -15$ .
Divide both sides: Divide both sides by $-2$ to solve for $x$ . $\frac{-2x}{-2} = \frac{-15}{-2}$ This simplifies to $x = 7.5$ .