Solve the Rational Equation for x (x+5)/5=6/(x-2)
The given problem is an algebraic equation involving rational expressions, which are fractions that have polynomials in their numerators and/or denominators. You are asked to find the value(s) of the variable x that make the equation true, essentially finding the common value(s) that satisfy both sides of the equation once simplified. This process will likely involve finding a common denominator, multiplying both sides to clear the fractions, and then solving for x like a typical algebraic equation.
$\frac{x + 5}{5} = \frac{6}{x - 2}$
Answer
Solution:
Step:1
Cross-multiply the two fractions to eliminate the denominators. This involves multiplying the numerator of the first fraction by the denominator of the second and vice versa. Write the equation as $(x + 5)(x - 2) = 5 \cdot 6$.
Step:2
Begin solving for $x$ by expanding and simplifying the equation.
Step:2.1
First, expand the product $(x + 5)(x - 2)$.
Step:2.1.1
Use the FOIL Method to expand the binomials.
Step:2.1.1.1
Distribute each term in the first binomial across the second binomial: $x(x - 2) + 5(x - 2) = 5 \cdot 6$.
Step:2.1.1.2
Distribute $x$ across $(x - 2)$: $x^2 - 2x + 5(x - 2) = 5 \cdot 6$.
Step:2.1.1.3
Continue distribution: $x^2 - 2x + 5x - 10 = 5 \cdot 6$.
Step:2.1.2
Combine like terms in the equation.
Step:2.1.2.1
Simplify each term in the equation.
Step:2.1.2.1.1
Square $x$: $x^2 - 2x + 5x - 10 = 5 \cdot 6$.
Step:2.1.2.1.2
Combine the $x$ terms: $x^2 + 3x - 10 = 5 \cdot 6$.
Step:2.1.2.1.3
Multiply $5$ by $-2$: $x^2 + 3x - 10 = 5 \cdot 6$.
Step:2.1.2.2
Add the $x$ coefficients: $x^2 + 3x - 10 = 5 \cdot 6$.
Step:2.2
Calculate the product of $5$ and $6$: $x^2 + 3x - 10 = 30$.
Step:2.3
Subtract $30$ from both sides to set the equation to zero: $x^2 + 3x - 40 = 0$.
Step:2.4
Factor the quadratic equation $x^2 + 3x - 40$.
Step:2.5
Use the AC method to factor the quadratic.
Step:2.5.1
Find two numbers that multiply to $-40$ and add to $3$: $-5$ and $8$.
Step:2.5.2
Write the equation in factored form: $(x - 5)(x + 8) = 0$.
Step:2.6
Apply the zero-product property, which states that if a product equals zero, then at least one of the factors must be zero.
Step:2.7
Solve for $x$ when the first factor equals zero.
Step:2.7.1
Set the first factor equal to zero: $x - 5 = 0$.
Step:2.7.2
Solve for $x$: $x = 5$.
Step:2.8
Solve for $x$ when the second factor equals zero.
Step:2.8.1
Set the second factor equal to zero: $x + 8 = 0$.
Step:2.8.2
Solve for $x$: $x = -8$.
Step:2.9
Combine the solutions to find all values of $x$ that satisfy the equation: $x = 5, -8$.
Knowledge Notes:
To solve a rational equation where the variables are in the numerator and the denominator, we often use cross-multiplication to eliminate the fractions. This is possible when we have an equation of the form $\frac{a}{b} = \frac{c}{d}$, which can be rewritten as $a \cdot d = b \cdot c$.
The FOIL Method stands for First, Outer, Inner, Last, and is used to expand the product of two binomials. The distributive property is applied to multiply each term in the first binomial by each term in the second.
Combining like terms is a process of adding or subtracting terms that have the same variable raised to the same power.
The zero-product property is used to solve quadratic equations that have been factored into the form $(x - p)(x - q) = 0$. It states that if the product of two factors is zero, then at least one of the factors must be zero.
Factoring is a method used to solve quadratic equations by expressing them as a product of two binomials. The AC method is one of the techniques used for factoring trinomials of the form $x^2 + bx + c$. It involves finding two numbers that multiply to the product of the coefficient of $x^2$ and the constant term (AC) and add to the coefficient of $x$ (B).
Finally, the solutions to the equation are the values of $x$ that make each factor equal to zero. These are the solutions to the original rational equation.
