Question

solve the system by graphing
\\(\

$$\begin{cases} 5x + 3y = 15 \\\\ 10x = -6y + 30 \\end{cases}$$

\\)

use the graphing tool to graph the system.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. there is exactly one solution. the solution set is \\(\\{\\ \\}\\).
(simplify your answer. type an ordered pair.)
\\(\bigcirc\\) b. there are infinitely many solutions. the solution set is \\(\\{(x,y) \mid 5x + 3y = 15\\}\\) or \\(\\{(x,y) \mid 10x = -6y + 30\\}\\).
\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Rewrite the second equation

Rewrite \(10x = -6y + 30\) to slope - intercept form (\(y=mx + b\)) or standard form. Let's rewrite both equations in slope - intercept form.

For the first equation \(5x+3y = 15\):
Subtract \(5x\) from both sides: \(3y=-5x + 15\)
Divide both sides by 3: \(y=-\frac{5}{3}x + 5\)

For the second equation \(10x=-6y + 30\):
Add \(6y\) to both sides: \(6y+10x = 30\)
Subtract \(10x\) from both sides: \(6y=-10x + 30\)
Divide both sides by 6: \(y=-\frac{10}{6}x + 5=-\frac{5}{3}x + 5\)

Step2: Analyze the two equations

We can see that both equations have the same slope (\(m =-\frac{5}{3}\)) and the same y - intercept (\(b = 5\)). This means that the two lines are coincident (they lie on top of each other).

Answer:

B. There are infinitely many solutions. The solution set is \(\{(x,y)\mid5x + 3y=15\}\) or \(\{(x,y)\mid10x=-6y + 30\}\).