Write the word or phrase that best completes each statement or answers the question.
-$$\begin{array} { l }
3 x + 6 y = 10 \
9 x + 18 y = 30
\end{array}$$
A) $( 0,4 )$
B) $( 6,1 )$
C) Dependent, $\left( x , \frac { 10 - 3 x } { 6 } \right)$
D) Inconsistent, ø

Question

Write the word or phrase that best completes each statement or answers the question.
-$$\begin{array} { l } 
3 x + 6 y = 10 \
9 x + 18 y = 30
\end{array}$$
A) $( 0,4 )$
B) $( 6,1 )$
C) Dependent, $\left( x , \frac { 10 - 3 x } { 6 } \right)$
D) Inconsistent, ø

Quizplus · Accepted Answer

By simplifying the first equation, we get 
$$3x+6y=10 \quad\Rightarrow\quad x+2y=\frac{10}{3}$$
By simplifying the second equation, we get
$$9x+18y=30 \quad\Rightarrow\quad x+2y=\frac{10}{3}$$
Notice that we end up with the same expression for both equations. This implies that the equations are dependent, meaning that one equation is a multiple of the other. Therefore, we can solve for one variable in terms of the other by choosing either equation. Let's choose the first equation:
$$x+2y=\frac{10}{3} \quad\Rightarrow\quad x=\frac{10}{3}-2y=\frac{10-6y}{3}$$
Thus, the solution to the system is $\left(x,\frac{10-3x}{6}\right)$. To check, we can substitute these values into both equations and see that they are satisfied. For example, using the second equation:
$$9x+18y=30 \quad\Rightarrow\quad 9\left(\frac{10-6y}{3}\right)+18y=30 \quad\Rightarrow\quad 10-6y+2(18y)=30$$
Simplifying, we get $y=4$. Substituting this back into the equation for $x$, we get $x=\frac{10-3(4)}{6}=1$. Thus, the solution is $\left(1,4\right)$, which matches choice (C).

Write the Word or Phrase That Best Completes Each Statement 3x+6y=109x+18y=30\begin{array} { l } 3 x + 6 y = 10 \\9 x + 18 y = 30\end{array}3x+6y=109x+18y=30​

Write the Word or Phrase That Best Completes Each Statement $\begin{array} { l } 3 x + 6 y = 10 \\9 x + 18 y = 30\end{array}$