{:[4y-3x=40],[4y=3x-30]:} Which of the following accurately describes all solutions to the system of equations shown? Choose 1 answer: (A) x=0 and y=0 (B) x=(5)/(3) and y=(45)/(4) (C) There are infinite solutions to the system. (D) There are no solutions to the system.

Analyze Given Equations: Let's analyze the system of equations given: \[ \begin{cases} 4y - 3x = 40 \\ 4y = 3x - 30 \end{cases} \] We can see that the two equations are supposed to represent the same line since they are both in terms of $y$. To check if they are indeed the same, we can try to manipulate one equation to see if it can be made to look like the other.Isolate y in First Equation: Let's isolate $y$ in the first equation: \[ 4y - 3x = 40 \] \[ 4y = 3x + 40 \] Now, we divide by 4 to solve for $y$: \[ y = \frac{3x}{4} + 10 \]Compare with Second Equation: Now let's compare this with the second equation: \[ 4y = 3x - 30 \] Dividing by 4 to solve for $y$, we get: \[ y = \frac{3x}{4} - \frac{30}{4} \] \[ y = \frac{3x}{4} - 7.5 \]Determine Consistency: We can see that the two equations: \[ y = \frac{3x}{4} + 10 \] and \[ y = \frac{3x}{4} - 7.5 \] cannot be the same because they have different constant terms. Therefore, the system of equations does not have infinite solutions, and since they are supposed to represent the same line, this inconsistency means there are no solutions to the system.

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{:[4y-3x=40],[4y=3x-30]:}
Which of the following accurately describes all solutions to the system of equations shown?
Choose 1 answer:
(A)
x=0 and
y=0
(B)
x=(5)/(3) and
y=(45)/(4)
(C) There are infinite solutions to the system.
(D) There are no solutions to the system.

$\begin{aligned} 4 y-3 x & =40 \\ 4 y & =3 x-30 \end{aligned}$ $\newline$ Which of the following accurately describes all solutions to the system of equations shown? $\newline$ Choose $1$ answer: $\newline$ (A) $x=0$ and $y=0$ $\newline$ (B) $x=\frac{5}{3}$ and $y=\frac{45}{4}$ $\newline$ (C) There are infinite solutions to the system. $\newline$ (D) There are no solutions to the system.

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Algebra 1

Write two-variable inequalities: word problems

Full solution

\begin{aligned} 4 y-3 x & =40 \\ 4 y & =3 x-30 \end{aligned}

\newline

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

\newline

Choose

1

answer:

\newline

(A)

x=0

and

y=0

\newline

(B)

x=\frac{5}{3}

and

y=\frac{45}{4}

\newline

\newline

(D) There are no solutions to the system.

Analyze Given Equations: Let's analyze the system of equations given: $\newline$ $\begin{cases} 4y - 3x = 40 \\ 4y = 3x - 30 \end{cases}$ $\newline$ We can see that the two equations are supposed to represent the same line since they are both in terms of $y$ . To check if they are indeed the same, we can try to manipulate one equation to see if it can be made to look like the other.
Isolate y in First Equation: Let's isolate $y$ in the first equation: $\newline$ $4y - 3x = 40$ $\newline$ $4y = 3x + 40$ $\newline$ Now, we divide by $4$ to solve for $y$ : $\newline$ $y = \frac{3x}{4} + 10$
Compare with Second Equation: Now let's compare this with the second equation: $\newline$ $4y = 3x - 30$ $\newline$ Dividing by $4$ to solve for $y$ , we get: $\newline$ $y = \frac{3x}{4} - \frac{30}{4}$ $\newline$ $y = \frac{3x}{4} - 7.5$
Determine Consistency: We can see that the two equations: $\newline$ $y = \frac{3x}{4} + 10$ $\newline$ and $\newline$ $y = \frac{3x}{4} - 7.5$ $\newline$ cannot be the same because they have different constant terms. Therefore, the system of equations does not have infinite solutions, and since they are supposed to represent the same line, this inconsistency means there are no solutions to the system.