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{:[t-5=(1)/(3)(u+15)-5],[t=(2)/(3)(u+9)]:}
Which of the following accurately describes all solutions to the system of equations shown?
Choose 1 answer:
(A)
u=-18 and
t=-6
(B)
u=-3 and
t=4
(c) There are infinite solutions to the system.
(D) There are no solutions to the system.

$\begin{aligned} t-5 & =\frac{1}{3}(u+15)-5 \\ t & =\frac{2}{3}(u+9) \end{aligned}$ $\newline$ Which of the following accurately describes all solutions to the system of equations shown? $\newline$ Choose $1$ answer: $\newline$ (A) $u=-18$ and $t=-6$ $\newline$ (B) $u=-3$ and $t=4$ $\newline$ (C) There are infinite solutions to the system. $\newline$ (D) There are no solutions to the system.

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Solve rational equations

Full solution

Q.

\begin{aligned} t-5 & =\frac{1}{3}(u+15)-5 \\ t & =\frac{2}{3}(u+9) \end{aligned}

\newline

Which of the following accurately describes all solutions to the system of equations shown?

\newline

Choose

1

answer:

\newline

(A)

u=-18

and

t=-6

\newline

(B)

u=-3

and

t=4

\newline

(C) There are infinite solutions to the system.

\newline

(D) There are no solutions to the system.

Write Equations: Write down the system of equations. $\newline$ We have the following system of equations: $\newline$ $t - 5 = \frac{1}{3}(u + 15) - 5$ $\newline$ $t = \frac{2}{3}(u + 9)$
Simplify First Equation: Simplify the first equation. $\newline$ $t - 5 = \frac{1}{3}(u + 15) - 5$ $\newline$ Multiply both sides by $3$ to clear the fraction: $\newline$ $3(t - 5) = u + 15 - 15$ $\newline$ $3t - 15 = u$
Substitute and Simplify: Substitute the expression for $u$ from the first equation into the second equation. $t = \left(\frac{2}{3}\right)(u + 9)$ $t = \left(\frac{2}{3}\right)(3t - 15 + 9)$ $t = \left(\frac{2}{3}\right)(3t - 6)$
Simplify Second Equation: Simplify the second equation. $\newline$ $t = \frac{2}{3}(3t - 6)$ $\newline$ Multiply both sides by $3$ to clear the fraction: $\newline$ $3t = 2(3t - 6)$ $\newline$ $3t = 6t - 12$
Solve for t: Solve for t. $\newline$ $3t = 6t - 12$ $\newline$ Subtract $6t$ from both sides: $\newline$ $-3t = -12$ $\newline$ Divide both sides by $-3$ : $\newline$ $t = 4$
Substitute for $u$ : Substitute the value of $t$ back into the first equation to solve for $u$ . $3t - 15 = u$ $3(4) - 15 = u$ $12 - 15 = u$ $u = -3$
Check Solution: Check the solution by substituting the values of $u$ and $t$ into both original equations. $\newline$ First equation: $t - 5 = \frac{1}{3}(u + 15) - 5$ $\newline$ $4 - 5 = \frac{1}{3}(-3 + 15) - 5$ $\newline$ $-1 = \frac{1}{3}(12) - 5$ $\newline$ $-1 = 4 - 5$ $\newline$ $-1 = -1$ (True) $\newline$ Second equation: $t = \frac{2}{3}(u + 9)$ $\newline$ $4 = \frac{2}{3}(-3 + 9)$ $\newline$ $4 = \frac{2}{3}(6)$ $\newline$ $t$ $0$ (True)

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