Algebraic Equation Setup
Let the unknown number be represented by the variable \(x\).
The problem translates to the following algebraic equation:
Twice the number plus 7 times its reciprocal equals 15.
This can be written as:
\(2x + 7 \times \frac{1}{x} = 15\)
Solving the Equation
To solve for \(x\), we first simplify the equation:
\(2x + \frac{7}{x} = 15\)
Assuming \(x\) is not zero, multiply every term by \(x\) to eliminate the fraction:
\(x \left( 2x + \frac{7}{x} \right) = 15x\)
\(2x^2 + 7 = 15x\)
Rearrange the terms to form a standard quadratic equation:
\(2x^2 - 15x + 7 = 0\)
Now, we can solve this quadratic equation by factoring:
We look for two numbers that multiply to \((2 \times 7 = 14)\) and add up to \(-15\). These numbers are \(-14\) and \(-1\).
Split the middle term:
\(2x^2 - 14x - x + 7 = 0\)
Factor by grouping:
\(2x(x - 7) - 1(x - 7) = 0\)
Factor out the common binomial \((x - 7)\):
\((2x - 1)(x - 7) = 0\)
Set each factor equal to zero to find the possible values of \(x\):
- \(2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}\)
- \(x - 7 = 0 \implies x = 7\)
Identifying the Number
The possible values for the number are \(\frac{1}{2}\) and $7$.
Comparing these values with the given options, the number $7$ is one of the choices.
Important Questions from Linear Equation in 1 Variable
A man has equal number of five, ten and twenty rupee notes amounting to Rs. 385. Find the total number of notes?
The sum of three consecutive number is 126. Find the highest number?
Simplify 5x(x + 2) + 4x
A.5x 2+ 10
B.9x + 10
C.5x 2- 14x
D.5x 2+ 14xSolve:
x - 4 = -3
A. 7
B. -1
C. -7
D. 1
If 4(3x - 2) = 2(3x + 8), Then x = ?
A. 1
B. 2
C. 3
D. 4
