$\begin{array}{l}\text { 55. } \frac{x-6}{x^{2}-5 x+6}=\frac{A}{x-2}+\frac{B}{x-3} \\ \Rightarrow 5 A-B=? \\\end{array}$

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Algebra 2

Simplify variable expressions using properties

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\begin{array}{l}\text { 55. } \frac{x-6}{x^{2}-5 x+6}=\frac{A}{x-2}+\frac{B}{x-3} \\ \Rightarrow 5 A-B=? \\\end{array}

Decompose Fraction: First, let's find the values of $A$ and $B$ by decomposing the fraction $\frac{x-6}{x^2-5x+6}$ into partial fractions.
Multiply by Denominator: We have $(x-6)/(x^2-5x+6) = A/(x-2) + B/(x-3)$ . To find $A$ and $B$ , we'll multiply both sides by the denominator $(x^2-5x+6)$ .
Expand and Group Terms: So, $(x-6) = A(x-3) + B(x-2)$ . Now we'll expand the right side.
Compare Coefficients: We get $x - 6 = Ax - 3A + Bx - 2B$ . Now, let's group like terms.
Solve Equations: This gives us $x - 6 = (A + B)x - (3A + 2B)$ . Now we can compare coefficients from both sides of the equation.
Substitute and Simplify: For the $x$ terms, we have $1 = A + B$ . And for the constant terms, we have $-6 = -3A - 2B$ .
Correct Mistake: Let's solve these two equations. From $1 = A + B$ , we can say $A = 1 - B$ .
Correct Mistake: Let's solve these two equations. From $1 = A + B$ , we can say $A = 1 - B$ .Substitute $A = 1 - B$ into $-6 = -3A - 2B$ to get $-6 = -3(1 - B) - 2B$ .
Correct Mistake: Let's solve these two equations. From $1 = A + B$ , we can say $A = 1 - B$ .Substitute $A = 1 - B$ into $-6 = -3A - 2B$ to get $-6 = -3(1 - B) - 2B$ .Simplify to get $-6 = -3 + 3B - 2B$ . This simplifies to $-6 = -3 + B$ .
Correct Mistake: Let's solve these two equations. From $1 = A + B$ , we can say $A = 1 - B$ .Substitute $A = 1 - B$ into $-6 = -3A - 2B$ to get $-6 = -3(1 - B) - 2B$ .Simplify to get $-6 = -3 + 3B - 2B$ . This simplifies to $-6 = -3 + B$ .Add $3$ to both sides to get $B = -3$ . But wait, there's a mistake here. We should have added $3$ to both sides to get $A = 1 - B$ $0$ .