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{:[11 e-9f+1=5(f-3e)],[2f+4e=3]:}
Consider the system of equations. Which of the following statements is true?
Choose 1 answer:
(A) There is only one solution
(e,f) and
e*f is positive.
(B) There is only one solution
(e,f) and
e*f is negative.
(c) There are infinitely many solutions.
(D) There are no solutions.

$11 e-9 f+1=5(f-3 e)$ $\newline$ $2 f+4 e=3$ $\newline$ Consider the system of equations. Which of the following statements is true? $\newline$ Choose $1$ answer: $\newline$ (A) There is only one solution $(e, f)$ and $e \cdot f$ is positive. $\newline$ (B) There is only one solution $(e, f)$ and $e \cdot f$ is negative. $\newline$ (C) There are infinitely many solutions. $\newline$ (D) There are no solutions.

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Solve rational equations

Full solution

Q.

11 e-9 f+1=5(f-3 e)

\newline

2 f+4 e=3

\newline

Consider the system of equations. Which of the following statements is true?

\newline

Choose

1

answer:

\newline

(A) There is only one solution

(e, f)

and

e \cdot f

is positive.

\newline

(B) There is only one solution

(e, f)

and

e \cdot f

is negative.

\newline

(C) There are infinitely many solutions.

\newline

(D) There are no solutions.

Write system of equations: Write down the system of equations. $\newline$ We have the following system of equations: $\newline$ $1$ ) $11e - 9f + 1 = 5(f - 3e)$ $\newline$ $2$ ) $2f + 4e = 3$
Simplify first equation: Simplify the first equation. $\newline$ Expand the right side of the first equation: $\newline$ $11e - 9f + 1 = 5f - 15e$ $\newline$ Now, move all terms involving variables to one side and constants to the other side: $\newline$ $11e + 15e = 5f + 9f - 1$ $\newline$ Combine like terms: $\newline$ $26e = 14f - 1$
Express $e$ in terms of $f$ : Express $e$ in terms of $f$ . $\newline$ Divide both sides by $26$ to solve for $e$ : $\newline$ $e = \frac{14f - 1}{26}$
Substitute expression for $e$ : Substitute the expression for $e$ into the second equation. $\newline$ Replace $e$ in the second equation with the expression found in Step $3$ : $\newline$ $2f + 4((14f - 1) / 26) = 3$
Simplify second equation: Simplify the second equation. $\newline$ Multiply through by $26$ to clear the fraction: $\newline$ $26(2f) + 4(14f - 1) = 26(3)$ $\newline$ $52f + 56f - 4 = 78$ $\newline$ Combine like terms: $\newline$ $108f - 4 = 78$
Solve for f: Solve for f. $\newline$ Add $4$ to both sides: $\newline$ $108f = 82$ $\newline$ Divide both sides by $108$ : $\newline$ $f = \frac{82}{108}$ $\newline$ Simplify the fraction: $\newline$ $f = \frac{41}{54}$
Substitute value of $f$ into $e$ : Substitute the value of $f$ back into the expression for $e$ . $e = \left(14\left(\frac{41}{54}\right) - 1\right) / 26$
Simplify expression to find $e$ : Simplify the expression to find $e$ . Multiply $14$ by $41/54$ : $e = (574/54 - 1) / 26$ Subtract $1$ (which is the same as subtracting $54/54$ ): $e = (520/54) / 26$ Simplify the fraction: $e = 20/54 / 26$ $e = 10/27$
Check solution in original equations: Check the solution in both original equations. $\newline$ Substitute $e = \frac{10}{27}$ and $f = \frac{41}{54}$ into the original equations to verify the solution: $\newline$ $1$ ) $11\left(\frac{10}{27}\right) - 9\left(\frac{41}{54}\right) + 1 \stackrel{?}{=} 5\left(\frac{41}{54} - 3\left(\frac{10}{27}\right)\right)$ $\newline$ $2$ ) $2\left(\frac{41}{54}\right) + 4\left(\frac{10}{27}\right) \stackrel{?}{=} 3$ $\newline$ Perform the calculations to check if both sides of the equations are equal.
Determine sign of $e*f$ : Determine the sign of $e*f$ . $\newline$ Since both $e$ and $f$ are positive fractions, their product will also be positive.

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